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Optimal access control for integrated services wireless networks
COMCOM 1451
Computer Communications 21 (1998) 1559–1570
Optimal access control for integrated services wireless networks Vittoria de Nitto Persone`*, Vincenzo Grassi Dipartimento di Informatica, Sistemi e Produzione, Universita` di Roma ‘‘Tor Vergata’’, via di Tor Vergata, 00133 Rome, Italy Received 30 September 1997; received in revised form 30 March 1998; accepted 28 April 1998
Abstract The requirement of ubiquitous information access motivates the development of wireless communication networks. Such networks are expected to support multimedia applications with different traffic characteristics and quality of service (QoS) requirements, similar to wired networks. Hence, in the case of wireless networks it is also important to devise effective resource allocation policies to guarantee a given QoS to different traffic classes. To increase frequencies reuse, a current trend in wireless cellular networks is toward microcellular architectures; this increases handoffs rates, with the possible consequent increase in the probability of connection interruptions. Hence, an important measure of the effectiveness of access control policies for wireless networks is the blocking probability of arriving connection requests in each cell. We consider the problem of optimal access control for a wireless network supporting multiple classes of traffic. In particular, we consider two optimization problems: minimizing any linear function of the blocking probabilities of different classes, and minimizing the blocking probability of one class, with a constraint on the blocking probability of the other class. For the first problem, we prove that the search for the optimal control policy can be limited to policies that base their decisions only on the occupancy levels. This result also implies that hysteresis-based policies are not optimal. For the second problem, we prove that within the class of fixed threshold policies a fractional threshold policy is optimal and provide a simple algorithm to calculate this threshold given the system parameters. 䉷 1998 Elsevier Science B.V. All rights reserved. Keywords: Integrated services network; Wireless network; Optimal access control; Blocking probability; Markov decision process
1. Introduction Ubiquitous information access and processing is a strong requirement in modern society, because of the increased flexibility and freedom this brings to individuals and organizations. This requirement has motivated the development of wireless communication infrastructures based on different technologies (e.g. cellular, indoor wireless LAN) [1, 2]. At the same time, wired networks are increasingly evolving according to a multimedia network model, because of the availability of multimedia user devices, and the gain in network management obtained by multiplexing multimedia connections. Hence, it is quite natural for users to expect a seamless transition from wired to wireless communication networks, which also means that the latter must be able to deal with multimedia traffic. Actually, next-generation wireless networks are designed according to this requirement [1, 3]. Multimedia traffic consists of several streams of traffic with different characteristics and quality of service (QoS) requirements. Hence, the main problem in multimedia * Corresponding author: e-mail: [email protected]
networks is how to satisfy these requirements by using policies for the control of access and the management of communication resources (e.g. buffers and bandwidth). This area has been the subject of intense study in the recent past, mainly in the framework of wired networks [4]. In this paper we address this problem in a wireless cellular framework. This means that proposed solutions must be evaluated by taking into consideration the peculiar characteristics of this environment (even if the results obtained are valid in a more general setting). In a cellular system, each cell makes available some amount of bandwidth (in the form of time slots, frequencies, spreading codes, or a combination of these): this amount may be fixed or change according to dynamic allocation schemes [2]. Two kinds of connection request compete for the use of this bandwidth: new connection requests originated within the cell, and handoff requests from already active connections entering the cell. Both of them may be rejected if the required QoS cannot be guaranteed. It should be noted that cellular networks are evolving into micro- or even pico-cellular networks to increase radio channel utilization [5], thus causing an increasing handoff traffic in each cell (and a consequent increase in the total arrival of
0140-3664/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S0140-366 4(98)00222-9
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connection requests in each cell). As a consequence, this can increase the likelihood of active connections interruption, caused by the impossibility of satisfying the required QoS in the cell they are entering. Hence an important goal of any bandwidth allocation policy in a cellular environment is the reduction of the blocking probability for arriving connection requests. The kind of QoS that must be guaranteed to a connection depends on the kind of traffic carried by that connection. Multimedia traffic can be broadly classified into two classes: realtime (voice, video) and non-realtime (data). The former has strong bandwidth and delay requirements, and some sort of resource reservation must be guaranteed to each connection that must be accommodated. On the other hand, the latter has much looser requirements. However, some minimal bandwidth should be guaranteed to each connection, to avoid excessive buffer overflows and consequent retransmissions, that would waste the already scarce available bandwidth. In this paper we focus our attention on the problem of optimal access control and bandwidth allocation in a cellular communication system, shared by different types of traffic. In particular, we consider a system where each cell has a fixed amount of bandwidth, shared by two types of traffic (class 1 and class 2), corresponding to constant bit rate (CBR) and available bit rate (ABR) traffic classes respectively [6]. Class 1 models traffic with stringent delay and bandwidth requirements (e.g. voice or video traffic), whereas class 2 models traffic with less stringent requirements (e.g. data traffic). Class 1 connections require a fixed amount of the system capacity and have preemptive priority over class 2 connections. On the other hand, class 2 connections equally share the total capacity not used by class 1 connections. It should be noted that, in the case of dynamic bandwidth allocation, our results concern the system management between two consecutive allocation changes, under the hypothesis that this time interval is long compared with a typical connection duration. For the control policies we consider, the control variable is the amount of the total bandwidth that can be used by class 1. This choice is motivated by the fact that class 1 connections are privileged with respect to class 2 connections, so that their access needs to be controlled to avoid ‘‘abuse’’ of this privilege. The objective function is based on the blocking probability for both classes. In particular, we analyze two optimization problems: (i) minimizing any linear function of the blocking probabilities of the two classes; (ii) minimizing the blocking probability of one class, with a constraint on the blocking probability of the other class. For both problems, we define a wide set of possible control policies, and obtain optimality results within this set. The set we consider for the first problem includes all the policies that base their decision on the occupancy level (number of users in the system) for the two classes. Moreover, it also includes policies that can use additional state
information besides the occupancy level. This additional information allows a policy to take different decisions under the same occupancy level. In this way we can also include in our search for optimal policies interesting policies that have been proposed in similar contexts for access control, like hysteresis-based policies [7, 8]. However, we prove that control policies that use additional state information cannot do better than policies that base their decisions only on the occupancy level for the two classes of users. In particular, this implies that hysteresisbased policies can be excluded from the search for optimal policies. For the second problem we prove that within the class of fixed threshold policies a fractional threshold policy is optimal, and provide a simple algorithm to calculate this threshold given the system parameters. The analysis of access control policies and resources management in the framework of wired networks has been the subject of several papers (see Ref. [4] and references cited therein). Some papers have also specifically addressed the same problem in the case of wireless cellular networks. In Ref. [9] a cellular network with two heterogeneous traffic classes is considered, and three different policies for access control are analyzed and numerically compared. In Ref. [10] a wireless network with a number of heterogeneous connections is considered, where the goal is to determine scheduling policies that achieve a given QoS (expressed in terms of the packet dropping probability for each connection); the region of achievable QoS and corresponding scheduling policies are derived. Analogously to Ref. [9], we also consider a cellular network with two heterogeneous classes of traffic; however, rather than numerically analyze some control policies, we obtain optimality results within a very large class of policies. This approach is similar, in its spirit, to the approach presented in Ref. [11]; we differ from that paper since we consider heterogenous traffic classes, whereas Ref. [11] considers two homogeneous traffic classes; moreover, we extend our analysis to a larger class of policies. The paper is organized as follows. In Section 2 we define formally a model of the system we are considering, and introduce the two considered objectives. In Section 3, we analyze the first objective: we define a very general class of control policies, and obtain optimality results by exploiting results from Markov decision theory. In Section 4 we analyze the second objective; also in this case we define a general class of control policies, and obtain results that allow us to define a simple algorithm that determines the optimal policy; we also present some numerical results obtained by using this algorithm. Section 5 concludes the paper.
2. The model We consider a cell in a cellular network with total available bandwidth equal to C that is shared by multimedia
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traffic entering the cell. In general, several classes of traffic can be identified in a multimedia environment. We focus our attention on two major classes: realtime and non-realtime. Their different QoS requirements result in different bandwidth requests. We assume that each realtime connection requires a fixed amount of bandwidth, thus being served according to a CBR service model; an arriving connection is dropped if the required bandwidth is not available. To reduce the dropping probability, we also assume that realtime connections have preemptive priority over non-realtime connections. On the other hand, non-realtime connections have less stringent requirements. In particular, they can tolerate variations in the service rate, thanks to flow control mechanism of the data transport protocols [12]. We assume that nonrealtime connections share equally the bandwidth not used by realtime connections, thus being served according to an ABR service model. This means that the service rate of each non-realtime connection can change over time, depending on the current number of realtime and non-realtime connections. Moreover, a threshold exists to the maximum number of accepted non-realtime connections, to guarantee some minimal bandwidth to each of them. Hence, we assume that an arriving non-realtime connection is dropped if the number of already present non-realtime connections is equal to the given threshold. More formally, we model the considered system as a service center with total service capacity (bandwidth) equal to C, where arriving users belong to two classes: class 1 (CBR) and class 2 (ABR). Users arrive at Poisson rate (l 1 and l 2, respectively) and their service demand is exponentially distributed with parameters m 1 and m 2 respectively. A class 1 user receives a fixed amount B1 ( ⬍ C) of bandwidth, and has preemptive priority over class 2 users. Hence, if n1 class 1 users are in the system, they use an amount n1B1 of the total capacity C. Class 2 users share bandwidth not used by class 1. Hence, if n2 class 2 users are in the system, each of them receives an amount (C ⫺ n1B1)/n2 of the available bandwidth. In the sequel, we assume, with no loss of generality, B1 1. Hence, the service completion rates for the two classes are n1m 1 and (C ⫺ n1)m 2 respectively. Both class 1 and 2 users may be rejected on their arrival. A class 1 user is rejected if the required bandwidth is not available. A class 2 user is rejected if the number of class 2 users already in the system is equal to a given threshold N2. In this paper we do not consider the case of queuing. Hence, a rejected user is lost. As explained in Section 1, it is important to manage the available bandwidth so as to maintain the blocking probability for both classes within acceptable limits. From the point of view of class 1, the best policy would of course be to make available to this class all the capacity C to satisfy arriving requests (note that, because of the preemptive priority, class 1 ‘‘does not see’’ class 2). But this policy could cause unacceptably high blocking probability for class 2.
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Hence, it is opportune to investigate access control policies that limit the access of class 1 connections, by limiting the total bandwidth amount that can be assigned to class 1. Two possible policies (already proposed in the literature in analogous contexts [13, 7, 8, 14]) are based on the fixed threshold and hysteresis concepts. In the fixed threshold policy, class 1 connections can use up to C 0 ⬍ C bandwidth with preemptive priority over class 2, while the remaining C ⫺ C 0 bandwidth is strictly reserved for class 2. The hysteresis policy has been proposed to get more flexibility with respect to a fixed threshold policy, by differentiating the policy behavior under normal load and overload system conditions. In our case, it may be defined by using two thresholds C 0 and C 00 , with 0 ⬍ C 0 ⬍ C 00 ⱕ C. The system starts in normal load condition and until the bandwidth used by class 1 is less than C 00 it remains in that condition. As soon as the bandwidth used by class 1 becomes equal to C 00 the system enters the overload condition, where only class 2 requests are served. The system returns in normal load condition only when the bandwidth used by class 1 drops below C 0 . Note that it may be C 00 C. Hence this policy can make available all the bandwidth to class 1 in normal load condition. Only if class 1 becomes too ‘‘greedy’’ is the system temporarily shut down for class 1, to give room to class 2. The effectiveness of these or other policies must be evaluated with respect to a given objective, defined in terms of the blocking probabilities for the two classes of users. Let PB1 and PB2 denote the blocking probabilities for class 1 and 2 respectively. In this paper we analyze the problem of determining optimal policies for the following objectives: (a) minimizing H·PB1 ⫹ L·PB2, with L, H ⱖ 0; (b1) minimizing PB1, subject to PB2 ⱕ P 00 ; (b2) minimizing PB2, subject to PB1 ⱕ P 0 . To this purpose, we define in the next sections two quite general classes of policies, and provide results concerning the above objectives.
3. Minimizing a linear objective function 3.1. Generalized hysteresis policies Let us consider the two policies described in Section 2. The fixed threshold policy bases its acceptance or rejection decision on the number of present class 1 users. On the other hand, the hysteresis policy bases its decision on an additional state information (normal load or overload), besides the number of present users. We consider a class of access control policies that can be considered as a generalization of these policies, called generalized hysteresis policies (GHP), where each arriving user is accepted or rejected according to the following state-dependent rules.
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(
d i; j; s2
Fig. 1. State transitions for process Xd .
The set of possible system states is defined as E {i; j; s 兩 0 ⱕ i ⱕ C; 0 ⱕ j ⱕ N2 ; s 僆 {1; 2; …; S}}
a 2; s2 ; s2
if i ⬎ C 0
a 2; s2 ; s1
if i ⱕ C 0
(where we assume that label s1 denotes normal load state and label s2 denotes overload state). Both fixed threshold and hysteresis policies choose deterministically a single decision in each state on class 1 arrival or departure. In general, a policy d 僆 GHP class can choose probabilistically among different decisions. We denote by pd i; j; s; a x; s 0 ; s 00 the probability of decision a x; s 0 ; s 00 in state i; j; s under policy d . Of course, it must hold 2 S S X X X
pd i; j; s; a x; s 0 ; s 00 1
In each state [i, j, s], i represents the number of class 1 users, j the number of class 2 users, and s represents a state label. The inclusion of a state label in the state definition allows us to define policies that take different decisions under the same occupancy level i,j. The decision to be taken in each state [i, j, s], with i ⬍ C is basically whether or not to accept an incoming class 1 user. More precisely, the set A of possible decisions in each state [i, j, s] has cardinality 2S 2 and its elements are defined as follows:
Under a given policy d 僆 GHP class, the system state evolution is described by a continuous time Markov process Xd fXd t 兩 ⱖ 0g, with state space E. Fig. 1 shows the possible transitions from a generic state i; j; s 僆 E, where ( l1 pd i; j; s; a x; s 0 ; s 00 ifx 1 0 l1 x; s 0 ifx 2
a 1; s 0 ; s 00 ; s 0 ; s 00 僆 {1; 2; …; S} :
m1 x; s 00 im1 pd i; j; s; a x; s 0 ; s 00
an arriving class 1 user is accepted and causes a transition to state i ⫹ 1; j; s 0 ; whereas a class 1 departure causes a transition to state i ⫺ 1; j; s 00 ; a 2; s 0 ; s 00 ; s 0 ; s 00 僆 {1; 2; …; S} : an arriving class 1 user is rejected and causes a transition to state i; j; s 0 ; whereas a class 1 departure causes a transition to state i ⫺ 1; j; s 00 : Of course, in each state C; j; s only decisions a 2; s 0 ; s 00 are possible. Each policy d 僆 GHP class corresponds to a given choice of the decision to be taken in each state, i.e. it can be defined as a function d : E ! A. In particular, it is easy to realize that the fixed threshold and hysteresis policies are special instances of the GHP class, and correspond to the following decisions choices:
x1 s 0 1 s 00 1
m2 i C ⫺ im2 As it can be noted, the rates for class 1 are policy dependent, as stated. On the other hand, the departure rate for class 2 is defined according to the ‘‘sharing available bandwidth’’ rule, introduced in Section 2. Let us denote by pd i; j; s the steady-state probabilities of Xd . We have the following expressions for the blocking probabilities of classes 1 and 2, denoted by PB1 and PB2 respectively: X X pd e; a·pd e 1 PB1 ei;j;s aa 2;·;·
PB2
X
pd e
2
ei;j;s jN2
• fixed threshold policy: (
d i; j; s
a 1; s; s if i ⬍ C 0 a 2; s; s if i ⱖ C 0
• hysteresis policy: (
d i; j; s1
a 1; s1 ; s1 if i ⬍ C 00 ⫺ 1 a 1; s2 ; s1 if i ⱖ C 00 ⫺ 1
3.2. Optimal policy The definition of the GHP class allows us to formulate under a Markov decision theory framework the problem of determining within this class an optimal policy for the objective (a) introduced in Section 2, i.e. (a) minimizing H·PB1 ⫹ L·PB2, with L, H ⱖ 0. To this purpose, we introduce a cost function r : E × A ! R, where R is the set of real numbers, that
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defines the cost per unit time accumulated by process Xd : 8 0 if x 1 and j ⬍ N2 > > > > > > H if x 2 and j ⬍ N2 > > > : L ⫹ H if x 2 and j N2 Let us denote by z(d ) the average cost per unit time of process Xd , defined as ZT E r
Xd
t; d
Xd
t dt 0
3 z
d lim T!∞ T Note that, by definition, each policy in GHP is a stationary Markov decision policy, i.e. it depends only on the current state. To apply results from the Markov decision theory, we limit our attention to policies d 僆 GHP such that Xd is irreducible. Then we have [15] X X r
e; a·pd
e; a·pd
e
4 z
d e僆E a僆A
By using Eqs. (1) and (2) we can rewrite Eq. (4) as follows: X X L·pd e; a·pd e z d ei;j;s
aa 1;·;·
j N2 X ⫹ ei;j;s
⫹
X
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are fixed by the characteristics of the system. Hence, to possibly limit the cardinality of E and consequently the cost of finding the optimal policy, we can only act on parameter S, that gives the number of possible state labels used to define a control policy (e.g. we need S 1 label for the fixed threshold policy, and S 2 labels for the hysteresis policy). Small S values reduce computational costs, but could exclude potentially better policies; the opposite is true with high S values. In the following, we prove that the search for the optimal solution of problem (a) can be limited to the smallest possible subset of DGHP, i.e. the subset obtained by fixing S 1, thus considerably reducing the computational cost. To this purpose, we adopt a technique presented in Ref. [11]. Basically, the steps of this technique are as follows (we refer to Ref. [11] for mathematical details). Instead of considering the infinite horizon non-discounted problem given by Eq. (6), we will consider the corresponding finite horizon discounted problem, will prove some property of the solution of this problem, and will infer equivalent property for the solution of Eq. (6), thanks to relations between the two kinds of problem. Let us denote by ZTa i; j; s the average discounted cost accumulated in [0, T] by process Xd with cost rate function r(·), starting from state i; j; s 僆 E, i.e. ZTa i; j; s ZT
L ⫹ H·pd e; a·pd e
E
aa 2;·;·
0
e⫺at r Xd t; d Xd t dt 兩 Xd 0 i; j; s ;
a⬎0
7
j N2 X
X
ei;j;s
aa 2;·;·
H·pd e; a·pd e
j ⬍ N2 H·PB1 ⫹ L·PB2
5
Let us denote by d * the optimal policy such that z dⴱ min {z d 兩 d 僆 GHP}
6
From Eq. (5) it follows that d * represents the solution for problem (a). Hence, standard algorithms developed in the framework of Markov decision theory can be used to determine this solution [15, 16]. Actually, from theory, we know that d * belongs to the subset DGHP 傺 GHP consisting of deterministic policies only, that choose deterministically a single decision in each state (i.e. for each state i; j; s there exists a single action a x; s 0 ; s 00 such that pd i; j; s; a x; s 0 ; s 00 1). Note that both the fixed threshold and the hysteresis policies defined in Section 2 belong to this subset, and hence are candidates for being the optimal solution. However, it should be noted that the cost of any algorithm used to determine d * is roughly proportional to 兩E兩 3, where: 兩E兩 S·C·N2 [15]. In the above expression for the cardinality of E, C and N2
Hence, the finite horizon discounted problem consists in determining the policy in DGHP that minimizes Eq. (7). Now, we can discretize this problem by using the standard uniformization technique [15]. It is easy to realize that the uniformization rate can be defined as L l1 ⫹ l2 ⫹ C m1 (under the assumption m 1 ⱖ m 2; the complementary assumption m 1 ⬍ m 2 gives rise to the same final results, and will not be considered in the sequel). Let Z ⴱa k; i; j; s be the optimal cost for the k-step discretized problem, starting from state i; j; s 僆 E. Normalizing the transition rates with respect to L , the following equation holds: Z ⴱa k; i; j; s l1 min{{Z ⴱa k ⫺ 1; i ⫹ 1; j; s 0 兩 1 ⱕ s 0 ⱕ S} 傼 {H ⫹ Z ⴱa k ⫺ 1; i; j; s 0 兩 1 ⱕ s 0 ⱕ S}} ⫹ im1 min{Z ⴱa k ⫺ 1; i ⫺ 1; j; s 0 ( ⴱa Z k ⫺ 1; i; j ⫹ 1; s if j ⬍ N2 0 兩 1 ⱕ s ⱕ S} ⫹ l2 L ⫹ Z ⴱa k ⫺ 1; i; j; s if j N2 ⫹ I j C ⫺ im2 Z ⴱa k ⫺ 1; i; j ⫺ 1; s ⫹ C ⫺ i m1 ⫺ I jm2 Z ⴱa k ⫺ 1; i; j; s
8
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where I j
1 if j ⬎ 0 0
if j 0
The first term in Eq. (8) is the contribution to the cost of class 1 arrivals, where we have the option of accepting the user and changing the state label to some s 0 , or rejecting the user, paying a cost H and changing state label to some s 0 ; the second term is the contribution of a class 1 departure, where we have the option of changing the state label to some s 0 ; the third term is the contribution of class 2 arrivals, where a cost L is accumulated only in saturated states; the fourth term is the contribution of a class 2 departure if j ⬎ 0, and the indicator function I( j) takes into account the case j 0. Finally, the last term is a consequence of uniformization. Boundary conditions for class 1 arrivals and departures can be simply handled by assuming Z ⴱa k; C ⫹ 1; j; s ∞. The following theorem holds for the optimal cost function defined by Eq. (8).
from theory that the optimal policy for problem (a) is deterministic, i.e. it chooses a single action in each state with probability 1. In this section, we consider problems (b1) and (b2) of Section 2, defined as (b1) minimizing PB1, subject to PB2 ⱕ P 00 ; (b2) minimizing PB2, subject to PB1 ⱕ P 0 . For these problems we consider a class of policies that base their decisions only on the occupancy levels for the two classes and that also includes probabilistic policies. We show that the optimal policies are probabilistic and also give algorithms to determine these policies. 4.1. Generalized fixed threshold policies The policies we consider consists of threshold-based policies that use a fractional threshold value to determine the acceptance or rejection of a class 1 user. More formally, these state-dependent policies can be defined as follows. The system state space is defined as E {i; j 兩 0 ⱕ i ⱕ C; 0 ⱕ j ⱕ N2 }
ⴱa
0
ⴱa
00
Theorem 1. Z k; i; j; s Z k; i; j; s for any s 0 ; s 00 僆 {1; 2; …; S}
Proof. We prove the theorem by induction on k. The basic step is trivial since Z ⴱa 0; i; j; s 0 0 for any s. Let us assume the theorem is true for Z ⴱa k ⫺ 1; ·. By using this induction hypothesis, it is straightforward to see from Eq. (8) that the theorem also holds for Z ⴱa k; ·. QED According to the results reported in Ref. [11], the property proved in Theorem 1 also applies to zⴱ z dⴱ defined by Eq. (6), since the state space of Xd is finite and Xd is irreducible. Hence, Theorem 1 states that the best policy in DGHP (such that Xd is irreducible) that uses state labels does not behave better (with respect to objective (a) of the best policy that bases its decision only on the occupancy level. As a consequence, we can limit the search for the optimal policy for problem (a) to the subset of DGHP consisting of policies that use no state label in their state description, thus reducing by a factor S the cardinality of the state space and by a factor S 2 the cardinality of the set of actions. Finally, note that, with respect to the two policies described in Section 2, this means that the hysteresis policy (that needs two different state labels for its implementation) cannot do better than single label policies (i.e. with S 1). 4. Minimizing blocking probability with constraint In Section 3 we have shown that the optimal policy for problem (a) takes its decisions on the basis only of the occupancy levels for the two classes. Moreover, we know
In each state i; j, i represents the number of class 1 users and j the number of class 2 users. The decision to be taken in each state i; j, with i ⬍ C is whether to accept or not an incoming class 1 user. Hence the set A of possible decisions in each state i; j is: • a(1): an arriving class 1 user is accepted and causes a transition to state i ⫹ 1; j; • a(2): an arriving class 1 user is rejected and causes no state transition. Of course, in each state C; j only decision a(2) is possible. The class of policies we consider, called generalized fixed threshold policies (GFTP), consists of policies d dn;p , defined as follows: 8 a 1 if i ⬍ n > > > > > < a 1 with probability p; if i n dn;p i ; j > > a 2 with probability 1 ⫺ p; if i n > > > : a 2 if i ⬎ n As it can be noted, each policy d n,p 僆 GFTP can be considered as a fixed threshold policy with fractional threshold t, where the integer part of t is equal to n and the fractional part is equal to p. Let Xd;n;p be the continuous time Markov process that describes the system behavior under policy d n,p. In Fig. 2 we show the possible transitions from a generic state i; j, where 8 l1 if i ⬍ n > > < l1 i pl1 if i n > > : 0 otherwise
m2 i C ⫺ im2
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Hence, the theorem can be proved by proving separately that PB1(n, p) is a decreasing function of n, for p 0, and PB1(n, p) is a decreasing function of p, for a given n. Let us prove that PB1 n; 0 ⬎ PB1 n ⫹ 1; 0. By using Eq. (9), and by some simple algebra, this is equivalent to proving that nX ⫹1 j0
which is true, since it holds that
Fig. 2. State transitions for process Xd ,n,p.
Now, we show that the optimal policies for objectives (b1) and (b2) do belong properly to GFTP, in the sense that an integer threshold n does not suffice, in general, to solve these problems. To this purpose, in the following we first prove that PB1 is a decreasing function of t, whereas PB2 is an increasing function of t. Then, we use these results to argue that the optimal thresholds for problems (b1) and (b2) are fractional, and define simple algorithms to determine these thresholds. Theorem 2. PB1 is a decreasing function of t n ⫹ p, n ⬍ C, 0 ⱕ p ⱕ 1. Proof. Let us consider the policy d n,p. Remember that, because of the preemptive priority, class 1 users do not see class 2 users. Hence, the class 1 behavior under this policy is described by the birth-and-death process whose state diagram is shown in Fig. 3. Let us denote by p (i) the steady state probability of state i for this process. We have [17]
p i
ri p 0; i!
p n ⫹ 1 p 2
p 0 4
0ⱕiⱕn
rn⫹1 p 0 n ⫹ 1!
nX ⫹1 j0
nX ⫹1 n X rj rj rj 1⫹ 1⫹r j! j! j ⫹ 1! j1 j0
⬎1⫹
n r X rj n ⫹ 1 j0 j!
Now, we show that PB1 n; p is decreasing with respect to p. Differentiating Eq. (9) with respect to p we have " # 2 rn rn⫹1 rn⫹1 ⫹p PB1 n; p ⫺ 1 ⫺ p n! n ⫹ 1! n ⫹ 1! 2p 2
3⫺2 " # j n⫹1 n⫹1 n n X r r r r 5 ⫹ ⫹p ⫺ 4 j! n ⫹ 1! n ⫹ 1! n! j0 2
3⫺1 j n⫹1 n X r r 5 ⫹p 4 j! n ⫹ 1! j0 By some simple algebra, it is easy to see that the condition (2/2p)PB1(n, p) ⬍ 0 is equivalent to the condition n nX ⫹1 r X rj rj ⬍ n ⫹ 1 j0 j! j! j0
which is true, as proved above. QED
3⫺1
n X rj rn⫹1 5 ⫹p j! n ⫹ 1! j0
To prove that PB2 is an increasing function of t is somewhat more complicated, since class 2 users ‘‘see’’ class 1 users. We proceed as follows. Let us define the following cost function for the process Xd ,n,p shown in Fig. 2: ( 0 if j ⬍ N2 r i; j 1 if j N2
where
r
n rj r X rj ⬎ j! n ⫹ 1 j0 j!
l1 m1
Then, it is PB1 n; p 1 ⫺ pp n ⫹ p n ⫹ 1
9
Note that PB1(n, p) is a continuous function of p and n.
Fig. 3. Behavior of class 1 under policy d n,p.
Let zd ,n,p be the average cost per unit time for Xd ,n,p, defined as ZT E r Xd;n;p t dt 0 zd;n;p lim T!∞ T It is easy to see that PB2 n; p zd;n;p . Now, we can transform by uniformization Xd ,n,p into an equivalent discrete time process, with uniformization rate L l1 ⫹ l2 ⫹ C m1 (as
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in Section 3, we assume m 1 ⱖ m 2). Without loss of generality, let us assume L 1. Let zd;n;p k; i; j denote the average cost accumulated in k steps by the uniformized process obtained from Xd ,n,p, starting from state i; j. The following equation holds: zd;n;p k; i; j r i; j ⫹ l1 8 zd;n;p k ⫺ 1; i ⫹ 1; j > > < × pzd;n;p k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j > > : zd;n;p k ⫺ 1; i; j
( ⫹ l2 ×
zd;n;p k ⫺ 1; i; j ⫹ 1
if j ⬍ N2
zd;n;p k ⫺ 1; i; j
if j N2
if i ⬍ n if i n if i ⬎ n
⫹ I j C ⫺ im2 zd;n;p k ⫺ 1; i; j ⫺ 1 10
where I(j) is defined as in Section 3.2 and takes into account the case in which there are no departures of class 2 because j 0. Moreover, it also holds that zd;n;p
zd;n;p k; i; j lim k!∞ k
11
Now, we need to prove the following two lemmas. Lemma 1. for all k. Proof.
zd;n;0 k; i; j is monotonically increasing in j
See Appendix A.
Lemma 2. for all k.
Proof. We prove the theorem by proving separately that zd;n;0 k; i; j is an increasing function of n, for each k, and that zd;n;p k; i; j is an increasing function of p, for each k. Then, from Eq. (11) and PB2 zd;n;p the theorem is proved. Let us prove by induction that zd;n⫹1;0 k; i; j ⫺ zd;n;0 k; i; j ⱖ 0. The basic step is trivially true since zd;n;0 0; i; j 0. Let us assume the theorem is true for k ⫺ 1. Let us consider zd;n;0 k; i; j. We have zd;n⫹1;0 k; i; j ⫺ zd;n;0 k; i; j l1 8 zd;n⫹1;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i ⫹ 1; j if i ⬍ n > > < if i n × zd;n⫹1;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j > > : zd;n⫹1;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i; j if i ⬎ n
⫹ im1 zd;n;p k ⫺ 1; i ⫺ 1; j
⫹ C ⫺ i m1 ⫺ I jm2 zd;n;p k ⫺ 1; i; j
Theorem 3. PB2 is an increasing function of t n ⫹ p, n ⬍ C, 0 ⱕ p ⱕ 1.
zd;n;0 k; i; j is monotonically increasing in i
Proof. See Appendix A. By using the two lemmas, we can prove the following theorem.
⫹l 2
( ×
zd;n⫹1;0 k ⫺ 1; i; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i; j ⫹ 1
if j ⬍ N2
zd;n⫹1;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i; j
if j N2
⫹im1 zd;n⫹1;0 k ⫺ 1; i ⫺ 1; j ⫺ zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹I j C ⫺ im2 zd;n⫹1;0 k ⫺ 1; i; j ⫺ 1 ⫺zd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ C ⫺ i m1 ⫺ I jm2 zd;n⫹1;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i; j
All the differences in the right-hand side of the above equation are ⱖ 0 by induction hypothesis, with the exception of the second difference in the first term (case i n); for this difference we have zd;n⫹1;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j ⱖ zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j by induction hypothesis ⱖ 0 by Lemma 2
Now, let us prove by induction that zd;n;p k; i; j ⫺ zd;n;q k; i; j ⱖ 0 for 0 ⱕ q ⬍ p ⱕ 1. Again, the basic step is trivially true. Then, we have
zd;n;p k; i; j ⫺ zd;n;q k; i; j 8 if i ⬍ n zd;n;p k ⫺ 1; i ⫹ 1; j ⫺ zd;n;q k ⫺ 1; i ⫹ 1; j > > < l1 × pzd;n;p k ⫺ 1; i ⫹ 1; j ⫺ qzd;n;q k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j ⫺ 1 ⫺ qzd;n;q k ⫺ 1; i; j if i n > > : zd;n;p k ⫺ 1; i; j ⫺ zd;n;q k ⫺ 1; i; j if i ⬎ n ( zd;n;p k ⫺ 1; i; j ⫹ 1 ⫺ zd;n;q k ⫺ 1; i; j ⫹ 1 if j ⬍ N2 ⫹ im1 zd;n;p k ⫺ 1; i ⫺ 1; j ⫺ zd;n;q k ⫺ 1; i ⫺ 1; j ⫹l2 × zd;n;p k ⫺ 1; i; j ⫺ zd;n;q k ⫺ 1; i; j if j N2 ⫹I j C ⫺ im2 zd;n;p k ⫺ 1; i; j ⫺ 1 ⫺ zd;n;q k ⫺ 1; i; j ⫺ 1
⫹ C ⫺ i m1 ⫺ I jm2 zd;n;p k ⫺ 1; i; j ⫺ zd;n;q k ⫺ 1; i; j
V. de Nitto Persone`, V. Grassi / Computer Communications 21 (1998) 1559–1570
Fig. 4. Optimal threshold for problem (b1).
All the differences in the right-hand side of the above equation are ⱖ 0 by induction hypothesis, with the exception of the second difference in the first term (case i n); for this difference we have
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else begin t_inf: 0; t_sup: C; repeat t: (t_sup ⫹ t_inf)/2; if PBx(t) ⬍ P then begin t_inf: t; diff: t_sup ⫺ t end else begin t_sup: t; diff: t ⫺ t_inf end until diff ⱕ 1; return(t) end; end.
pzd;n;p k ⫺ 1; i ⫹ 1; j ⫺ qzd;n;q k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j ⫺ 1 ⫺ qzd;n;q k ⫺ 1; i; j ⱖ pzd;n;p k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j ⫺ qzd;n;p k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ qzd;n;p k ⫺ 1; i; j by induction hypothesis p ⫺ q zd;n;p k ⫺ 1; i ⫹ 1; j ⫺ zd;n;p k ⫺ 1; i; j ⱖ 0 by Lemma 2 and p ⬎ q
(Note that, actually, Lemma 2 has been proven for the special case p 0; the extension to the case p ⬎ 0 is trivial and is omitted.) QED Now, let us consider problem (b1): minimizing PB1, subject to PB2 ⱕ P 00 . From Theorems 2 and 3 we have that the behavior of PB1 and PB2 as a function of t is as depicted in Fig. 4. Hence, any value t ⱕ t00 (shown by the shadowed region) satisfies the constraint PB2 t ⱕ P 00 . Then, the optimal value for t is t t00 , since PB1 t is a decreasing function of t. This means that the optimal policy in the class GFTP for problem (b1) is the policy d1 dn 00 ;p 00 · such that PB2 n 00 ⫹ p 00 P 00
12
Analogously, the optimal policy in the class GFTP for problem (b2) (minimizing PB2, subject to PB1 ⱕ P 0 ), is the policy d2 dn 0 ;p 0 · such that: PB1 n 0 ⫹ p 0 P 0
13
It is easy to be convinced that the solution of Eqs. (12) and (13) is, in general, non-integer, thus proving our statement that the optimal threshold for problems (b1) and (b2) is fractional. In the following, we give a simple algorithm to determine the optimal thresholds t00 and t0 , based on a binary search of the solutions of Eqs. (12) and (13) respectively. The algorithm is guaranteed to converge to the solutions thanks to the monotonicity properties of PB2 and PB1. Algorithm B x; P; b begin if PBx(b) ⬎ P then no solution
The algorithm parameters are (x 2, P P 00 , b 0), and (x 1, P P 0 , b C) for problems (b1) and (b2) respectively, whereas 1 is the required resolution for the problem solution (e.g. 1 10 ⫺4). Note that in algorithm B 1; P 0 ; C, PB1(t) can be evaluated by using the closedform expression (9) in Theorem 2. On the other hand, no closed-form expression is available for the evaluation of PB2(t) in algorithm B 2; P 00 ; 0; we have to solve the balance equations of Markov process Xd;n;p (with n ⫹ p t) to calculate PB2(t). Hence, each step in the iterative part of algorithm B 2; P 00 ; 0 has a greater complexity than the corresponding step of algorithm B 1; P 0 ; C (in other words, solving problem (b1) costs more than solving problem (b2)). 4.2. Numerical results In Tables 1–4 we present some numerical results obtained by algorithm B x; P; b. Tables 1 and 2 refer to problem (b1), with constraint PB2 ⱕ 0.05; Tables 3 and 4 refer to problem (b2), with constraint PB1 ⱕ 0.01. In all the tables, column t_opt reports the optimal threshold calculated by the algorithm and columns PB1(t_opt) and PB2(t_opt) the corresponding blocking probabilities. Note that, by the algorithm definition, the value t_opt is such that the constraint value is exactly met by the corresponding blocking probability; hence, we have PB2(t_opt) 0.05 in Tables 1 and 2, and PB1(t_opt) 0.01 in Tables 3 and 4. However, the actual implementation of a fractional threshold may be somewhat more complex than the implementation of an integer threshold. As a consequence, an alternative choice (Choice 1) could be to use as threshold the closest integer that satisfies the constraint. In this case, it
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Table 1 Problem (b1): C 20, N2 20, B1 1, l 1 10, m 1 1, m 2 1 PB2 ⱕ 0.05
l2 10 11 12 14
t_opt
14.872 12.067 10.141 7.122
PB1(t_opt)
0.0390 0.1172 0.2062 0.3968
PB2(t_opt)
0.05 0.05 0.05 0.05
Choice 1 cost %PB1(t_inf)
Choice 2 cost %PB2(t_closest)
⫹ 45 ⫹2 ⫹4 ⫹3
⫹3 — — —
is useful to evaluate the ‘‘cost’’ of such a choice, which may be measured as the corresponding percentage increase of the non-constrained blocking probability. The fifth columns of Tables 1–4 (labeled ‘‘Choice 1 cost’’) shows this percentage increase when using as threshold the closest integer that satisfies the constraint. This integer is equal to bt_optc (t_inf) for Tables 1 and 2, whereas it is equal to dt_opte (t_sup) for Tables 3 and 4. As it can be noted, the percentage increase ranges from 0.2 to 45%, with values below 10% in most of the cases. These results suggest that in many cases this integer threshold may be satisfactory. However, it is worth pointing out that Choice 1, i.e. the use of the closest integer that satisfies the constraint, can yield a threshold that is quite far from the optimal one. This can be noted, for example, from the first row of Table 1 (where t_opt 14.872 and t_inf 14), or from Table 4 (where t_opt 16.2 and t_sup 17). It can also be noted that, in some of these cases, the cost of Choice 1 is quite high, as shown by the fifth column values. Thus, if an integer threshold is considered preferable for implementation simplicity, this suggests as alternative choice the use of the integer closest to the optimal threshold (Choice 2), even if this can lead to violating the given constraint. In this case, the ‘‘cost’’ of Choice 2 may be evaluated by measuring how much the constraint is violated. Hence, the last columns of the tables (labeled Choice 2 cost) show the impact of using the integer closest to the optimal fraction (t_closest). In all the tables, this last columns show the percentage increase of the constrained probability PBx with respect to the constraint value (0.05 for Tables 1 and 2, and 0.01 for Tables 3 and 4), only in the cases where the use of t_closest leads to a constraint violation (in the other cases, Choice 2 coincides with Choice 1). In the case of the first row of Table 1, we see that the cost of Choice 2 (value of %PB2(t_closest)) results equal to 3%.
Table 3 Problem (b2): C 20, N2 20, B1 1, l 2 10, m 1 1, m 2 1, PB1 ⱕ 0.01
l1
t_opt
PB1 (t_opt)
PB2(t_opt)
Choice 1 cost %PB2(t_sup)
Choice 2 cost %PB1(t_closest)
8 9 10 11 12
14.889 16.200 17.506 18.756 19.969
0.01 0.01 0.01 0.01 0.01
0.0118 0.0311 0.0689 0.1283 0.2055
⫹ 1.7 ⫹ 6.8 ⫹ 3.2 ⫹ 1.2 ⫹ 0.2
— ⫹ 10 — — —
This small cost may be weighed against the cost of Choice 1 (value of %PB1(t_inf)), that results equal to 45%. On the other hand, we see in Table 4 that the Choice 2 cost results equal to 10%, against a cost of Choice 1 (value of %PB2(t_sup)) that ranges from 3 to 28%. Finally, it is worth noting that when either Choice 1 or 2 is adopted, Algorithm B x; P; b is still useful to determine the corresponding integer threshold.
5. Conclusions We have analyzed the problem of access control in a wireless cellular environment that supports multiple classes of traffic with different QoS requirements. The class with more critical requirements needs to be privileged in resource assignment; however, to avoid abuse of this privilege it is opportune to implement some form of control on the access requests of the privileged class. Hence, we have studied the problem of determining optimal access control policies with respect to two different objectives, both based on the blocking probabilities for the considered classes of users. The blocking probability is an important metric to measure the effectiveness of control policies in a microcellular environment, where a large fraction of the incoming traffic consists of handoff requests of already active connections, whose interruptions should be avoided as much as possible, to avoid users dissatisfaction. For the first objective (minimizing a linear function of the blocking probabilities), we have defined a very general class of control policies, and we have shown that the optimal policy belongs to a small subset of this class, thus improving the efficiency of standard algorithms used to determine this policy. For the Table 4 Problem (b2): C 20, N2 20, B1 1, l 1 9, m 1 1, m 2 1, PB1 ⱕ 0.01
Table 2 Problem (b1): C 20, N2 20, B1 1, l 2 10, m 1 1, m 2 1, PB2 ⱕ 0.05
l1
t_opt
PB1(t_opt)
PB2(t_opt)
Choice 1 cost %PB1(t_inf)
11 12 13
13.239 12.463 12.033
0.1100 0.1765 0.2355
0.05 0.05 0.05
⫹7 ⫹ 12.5 ⫹ 0.8
Choice 2 cost %PB2(t_closest) — — —
l2
t_opt
PB2(t_opt)
PB2(t_opt)
Choice 1 cost %PB2(t_sup)
Choice 2 cost %PB1(t_closest)
7 8 9 10 11 12
16.200 16.200 16.200 16.200 16.200 16.200
0.01 0.01 0.01 0.01 0.01 0.01
0.000604 0.00317 0.0115 0.0311 0.0650 0.111
⫹ 28 ⫹ 18 ⫹ 11 ⫹ 6.8 ⫹ 3.4 ⫹3
⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10
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second objective (minimizing the blocking probability of a class, with a constraint on the blocking probability of the other class), we have proved results that have allowed us to define a simple algorithm that determines the optimal threshold for admission control. Since the resulting optimal threshold is fractional, we have also analyzed the impact of approximating this threshold by an integer threshold, that is simpler to implement.
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Acknowledgements This work has been partially supported by the CNR project ‘‘Ambienti avanzati per comunicazioni a pacchetto’’, and by MURST funds. We would like to thank the anonymous referees for their suggestions that helped us to improve the paper.
Appendix A
Lemma 1.
zd;n;0 k; i; j is monotonically increasing in j for all k.
Proof. We have to prove that zd;n;0 k; i; j ⫹ 1 ⱖ zd;n;0 k; i; j. We prove the lemma by induction on k. The basic step is trivial since zd;n;0 0; i; j 0 for any [i, j]. Let us assume the lemma is true for k ⫺ 1. We have zd;n;0 k; i; j ⫹ 1 ⫺ zd;n;0 k; i; j r i; j ⫹ 1 ⫺ r i; j ⫹ l1 ( zd;n;0 k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i ⫹ 1; j if i ⬍ n × if i ⱖ n zd;n;0 k ⫺ 1; i; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i; j ( zd;n;0 k ⫺ 1; i; j ⫹ 2 ⫺ zd;n;0 k ⫺ 1; i; j ⫹ 1 if j ⬍ N2 ⫺ 1 ⫹ l2 × 0 if j N2 ⫺ 1 ⫹ im1 zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹ C ⫺ im2 zd;n;0 k ⫺ 1; i; j ⫺ I jzd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ C ⫺ i m1 ⫺ m2 zd;n;0 k ⫺ 1; i; j ⫹ 1 ⫺ C ⫺ i m1 ⫺ I jm2 zd;n;0 k ⫺ 1; i; j ⱖ 0 since, by induction hypothesis, all the differences in the right-hand side of the above equation are ⱖ 0. QED
Lemma 2.
zd;n;0 k; i; j is monotonically increasing in i for all k.
Proof. We have to prove that zd;n;0 k; i ⫹ 1; j ⱖ zd;n;0 k; i; j. We prove the lemma by induction on k. The basic step is trivial since zd;n;0 0; i; j 0 for any [i, j]. Let us assume the lemma is true for k ⫺ 1. We have zd;n;0 k; i ⫹ 1; j ⫺ zd;n;0 k; i; j r i; j ⫹ 1 ⫺ r i; j ⫹ l1 8 zd;n;0 k ⫺ 1; i ⫹ 2; j ⫺ zd;n;0 k ⫺ 1; i ⫹ 1; j if i ⬍ n ⫺ 1 > > < if i n ⫺ 1 × 0 > > : if i ⬎ n ⫺ 1 zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j ( zd;n;0 k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i; j ⫹ 1 if j ⬍ N2 ⫹ l2 × zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j if j N2 ⫹ m1 i ⫹ 1zd;n;0 k ⫺ 1; i; j ⫺ izd;n;0 k ⫺ 1; i ⫺ 1; j ⫹ I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ 1 ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ m1 ⫺ I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j The first three terms in the right-hand side of the above equation are ⱖ 0 by definition of r(·) and by induction hypothesis. Let
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us consider the remaining three terms; we can rearrange them as follows:
m1 i ⫹ 1zd;n;0 k ⫺ 1; i; j ⫺ izd;n;0 k ⫺ 1; i ⫺ 1; j ⫹I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ 1 ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ m1 ⫺ I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j im1 zd;n;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹I j C ⫺ i ⫺ 1m2 zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ 1 ⫺ zd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ m1 ⫺ I jm2 C ⫺ i ⫺ 1 zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j ⫹m1 zd;n;0 k ⫺ 1; i; j ⫺ I jm2 zd;n;0 k ⫺ 1; i; j ⫺ 1 ⫺ m1 ⫺ I jm2 zd;n;0 k ⫺ 1; i; j ⱖ 0 by induction hypothesis and by Lemma 1. QED References [1] P. Agrawal, et al., SWAN: a mobile multimedia wireless network, IEEE Personal Communications 3 (2) (1997) 18–33. [2] V.O.K. Li, X. Qiu, Personal communication systems (PCS), Proceedings of the IEEE 83 (9) (1995) 1210–1243. [3] A. Allwan, et al., Adaptive mobile multimedia networks, IEEE Personal Communications 3 (2) (1997) 34–51. [4] D. Towsley, Providing quality of service in packet switched networks, in: L. Donatiello, R. Nelson (Eds.), Performance Evaluation of Computer and Communication Systems, Springer, 1993, pp. 560– 586. [5] W.C.Y. Lee, Smaller cells for greater performance, IEEE Communication Magazine, November, 1991. [6] A. Iwata, N. Mori, C. Ikeda, H. Suzuki, M. Ott, ATM connection and traffic management schemes for multimedia internetworking, Communications of the ACM 38 (2) (1995) 72–89. [7] D. McMillan, Delay analysis of a cellular mobile priority queuing system, IEEE/ACM Trans. on Networking 3 (3) (1995) 310–319. [8] R.G. Schehrer, On a cut-off priority queuing system with hysteresis and unlimited waiting room, Computer Networks and ISDN Systems 20 (1990) 45–56. [9] M. Naghshineh, A.S. Acampora, QoS provisioning in micro-cellular networks supporting multiple classes of traffic, Wireless Networks 2 (3) (1996) 195–204. [10] J.M. Capone, I. Stavrakakis, Achievable QoS and scheduling policies for integrated services wireless networks, Performance Evaluation 2788 (1996) 347–365. [11] R. Ramjee, D. Towsley, R. Nagarajan, On optimal call admission control in cellular networks, Wireless Networks 3 (1) (1997) 29–42. [12] D.E. Comer, D.L. Stevens, Internetworking with TCP/IP, Prentice Hall, 1984. [13] R. Guerin, Queuing-blocking system with two arrival streams and guard channels, IEEE Trans. on Communications 36 (2) (1988) 153–163.
[14] Q.-A. Zeng, K. Mukumoto, A. Fukuda, Performance analysis of mobile cellular radio system with priority reservation handoff procedures, Proc. of IEEE VTC 94 Conference, 1994, pp. 1829–1833. [15] D.P. Heyman, M.J. Sobel, Stochastic Models in Operations Research, McGraw-Hill, 1984. [16] S. Ross, Applied Probability Models with Optimization Applications, Holden-Day, 1970. [17] L. Kleinrock, Queuing Systems, vol. I: Theory, Wiley, 1976.
Vittoria de Nitto Persone` received the Laurea degree in Computer Science from the University of Pisa, Italy, in 1984. She is currently Assistant Professor at the Department of Computer Engineering, University of Roma ‘‘Tor Vergata’’, Roma, Italy. Her main research interests include performance modeling and evaluation, computer and communication systems, wireless systems, distributed systems and parallel processing. She has published several journal and conference papers and participated in the organizing and scientific committees of various international conferences. E-mail: [email protected], [email protected]
Vincenzo Grassi received the Laurea degree in Computer Science from the University of Pisa, Italy, in 1984. He is currently Associate Professor at the Department of Computer Engineering, University of Roma ‘‘Tor Vergata’’, Italy. His research interests include performance and reliability analysis of computer and communication systems, quality of service analysis in high speed and wireless networks, design of distributed and nomadic applications. Professor Grassi is a member of ACM. E-mail: [email protected], [email protected]
Computer Communications 21 (1998) 1559–1570
Optimal access control for integrated services wireless networks Vittoria de Nitto Persone`*, Vincenzo Grassi Dipartimento di Informatica, Sistemi e Produzione, Universita` di Roma ‘‘Tor Vergata’’, via di Tor Vergata, 00133 Rome, Italy Received 30 September 1997; received in revised form 30 March 1998; accepted 28 April 1998
Abstract The requirement of ubiquitous information access motivates the development of wireless communication networks. Such networks are expected to support multimedia applications with different traffic characteristics and quality of service (QoS) requirements, similar to wired networks. Hence, in the case of wireless networks it is also important to devise effective resource allocation policies to guarantee a given QoS to different traffic classes. To increase frequencies reuse, a current trend in wireless cellular networks is toward microcellular architectures; this increases handoffs rates, with the possible consequent increase in the probability of connection interruptions. Hence, an important measure of the effectiveness of access control policies for wireless networks is the blocking probability of arriving connection requests in each cell. We consider the problem of optimal access control for a wireless network supporting multiple classes of traffic. In particular, we consider two optimization problems: minimizing any linear function of the blocking probabilities of different classes, and minimizing the blocking probability of one class, with a constraint on the blocking probability of the other class. For the first problem, we prove that the search for the optimal control policy can be limited to policies that base their decisions only on the occupancy levels. This result also implies that hysteresis-based policies are not optimal. For the second problem, we prove that within the class of fixed threshold policies a fractional threshold policy is optimal and provide a simple algorithm to calculate this threshold given the system parameters. 䉷 1998 Elsevier Science B.V. All rights reserved. Keywords: Integrated services network; Wireless network; Optimal access control; Blocking probability; Markov decision process
1. Introduction Ubiquitous information access and processing is a strong requirement in modern society, because of the increased flexibility and freedom this brings to individuals and organizations. This requirement has motivated the development of wireless communication infrastructures based on different technologies (e.g. cellular, indoor wireless LAN) [1, 2]. At the same time, wired networks are increasingly evolving according to a multimedia network model, because of the availability of multimedia user devices, and the gain in network management obtained by multiplexing multimedia connections. Hence, it is quite natural for users to expect a seamless transition from wired to wireless communication networks, which also means that the latter must be able to deal with multimedia traffic. Actually, next-generation wireless networks are designed according to this requirement [1, 3]. Multimedia traffic consists of several streams of traffic with different characteristics and quality of service (QoS) requirements. Hence, the main problem in multimedia * Corresponding author: e-mail: [email protected]
networks is how to satisfy these requirements by using policies for the control of access and the management of communication resources (e.g. buffers and bandwidth). This area has been the subject of intense study in the recent past, mainly in the framework of wired networks [4]. In this paper we address this problem in a wireless cellular framework. This means that proposed solutions must be evaluated by taking into consideration the peculiar characteristics of this environment (even if the results obtained are valid in a more general setting). In a cellular system, each cell makes available some amount of bandwidth (in the form of time slots, frequencies, spreading codes, or a combination of these): this amount may be fixed or change according to dynamic allocation schemes [2]. Two kinds of connection request compete for the use of this bandwidth: new connection requests originated within the cell, and handoff requests from already active connections entering the cell. Both of them may be rejected if the required QoS cannot be guaranteed. It should be noted that cellular networks are evolving into micro- or even pico-cellular networks to increase radio channel utilization [5], thus causing an increasing handoff traffic in each cell (and a consequent increase in the total arrival of
0140-3664/98/$ - see front matter 䉷 1998 Elsevier Science B.V. All rights reserved. PII: S0140-366 4(98)00222-9
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connection requests in each cell). As a consequence, this can increase the likelihood of active connections interruption, caused by the impossibility of satisfying the required QoS in the cell they are entering. Hence an important goal of any bandwidth allocation policy in a cellular environment is the reduction of the blocking probability for arriving connection requests. The kind of QoS that must be guaranteed to a connection depends on the kind of traffic carried by that connection. Multimedia traffic can be broadly classified into two classes: realtime (voice, video) and non-realtime (data). The former has strong bandwidth and delay requirements, and some sort of resource reservation must be guaranteed to each connection that must be accommodated. On the other hand, the latter has much looser requirements. However, some minimal bandwidth should be guaranteed to each connection, to avoid excessive buffer overflows and consequent retransmissions, that would waste the already scarce available bandwidth. In this paper we focus our attention on the problem of optimal access control and bandwidth allocation in a cellular communication system, shared by different types of traffic. In particular, we consider a system where each cell has a fixed amount of bandwidth, shared by two types of traffic (class 1 and class 2), corresponding to constant bit rate (CBR) and available bit rate (ABR) traffic classes respectively [6]. Class 1 models traffic with stringent delay and bandwidth requirements (e.g. voice or video traffic), whereas class 2 models traffic with less stringent requirements (e.g. data traffic). Class 1 connections require a fixed amount of the system capacity and have preemptive priority over class 2 connections. On the other hand, class 2 connections equally share the total capacity not used by class 1 connections. It should be noted that, in the case of dynamic bandwidth allocation, our results concern the system management between two consecutive allocation changes, under the hypothesis that this time interval is long compared with a typical connection duration. For the control policies we consider, the control variable is the amount of the total bandwidth that can be used by class 1. This choice is motivated by the fact that class 1 connections are privileged with respect to class 2 connections, so that their access needs to be controlled to avoid ‘‘abuse’’ of this privilege. The objective function is based on the blocking probability for both classes. In particular, we analyze two optimization problems: (i) minimizing any linear function of the blocking probabilities of the two classes; (ii) minimizing the blocking probability of one class, with a constraint on the blocking probability of the other class. For both problems, we define a wide set of possible control policies, and obtain optimality results within this set. The set we consider for the first problem includes all the policies that base their decision on the occupancy level (number of users in the system) for the two classes. Moreover, it also includes policies that can use additional state
information besides the occupancy level. This additional information allows a policy to take different decisions under the same occupancy level. In this way we can also include in our search for optimal policies interesting policies that have been proposed in similar contexts for access control, like hysteresis-based policies [7, 8]. However, we prove that control policies that use additional state information cannot do better than policies that base their decisions only on the occupancy level for the two classes of users. In particular, this implies that hysteresisbased policies can be excluded from the search for optimal policies. For the second problem we prove that within the class of fixed threshold policies a fractional threshold policy is optimal, and provide a simple algorithm to calculate this threshold given the system parameters. The analysis of access control policies and resources management in the framework of wired networks has been the subject of several papers (see Ref. [4] and references cited therein). Some papers have also specifically addressed the same problem in the case of wireless cellular networks. In Ref. [9] a cellular network with two heterogeneous traffic classes is considered, and three different policies for access control are analyzed and numerically compared. In Ref. [10] a wireless network with a number of heterogeneous connections is considered, where the goal is to determine scheduling policies that achieve a given QoS (expressed in terms of the packet dropping probability for each connection); the region of achievable QoS and corresponding scheduling policies are derived. Analogously to Ref. [9], we also consider a cellular network with two heterogeneous classes of traffic; however, rather than numerically analyze some control policies, we obtain optimality results within a very large class of policies. This approach is similar, in its spirit, to the approach presented in Ref. [11]; we differ from that paper since we consider heterogenous traffic classes, whereas Ref. [11] considers two homogeneous traffic classes; moreover, we extend our analysis to a larger class of policies. The paper is organized as follows. In Section 2 we define formally a model of the system we are considering, and introduce the two considered objectives. In Section 3, we analyze the first objective: we define a very general class of control policies, and obtain optimality results by exploiting results from Markov decision theory. In Section 4 we analyze the second objective; also in this case we define a general class of control policies, and obtain results that allow us to define a simple algorithm that determines the optimal policy; we also present some numerical results obtained by using this algorithm. Section 5 concludes the paper.
2. The model We consider a cell in a cellular network with total available bandwidth equal to C that is shared by multimedia
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traffic entering the cell. In general, several classes of traffic can be identified in a multimedia environment. We focus our attention on two major classes: realtime and non-realtime. Their different QoS requirements result in different bandwidth requests. We assume that each realtime connection requires a fixed amount of bandwidth, thus being served according to a CBR service model; an arriving connection is dropped if the required bandwidth is not available. To reduce the dropping probability, we also assume that realtime connections have preemptive priority over non-realtime connections. On the other hand, non-realtime connections have less stringent requirements. In particular, they can tolerate variations in the service rate, thanks to flow control mechanism of the data transport protocols [12]. We assume that nonrealtime connections share equally the bandwidth not used by realtime connections, thus being served according to an ABR service model. This means that the service rate of each non-realtime connection can change over time, depending on the current number of realtime and non-realtime connections. Moreover, a threshold exists to the maximum number of accepted non-realtime connections, to guarantee some minimal bandwidth to each of them. Hence, we assume that an arriving non-realtime connection is dropped if the number of already present non-realtime connections is equal to the given threshold. More formally, we model the considered system as a service center with total service capacity (bandwidth) equal to C, where arriving users belong to two classes: class 1 (CBR) and class 2 (ABR). Users arrive at Poisson rate (l 1 and l 2, respectively) and their service demand is exponentially distributed with parameters m 1 and m 2 respectively. A class 1 user receives a fixed amount B1 ( ⬍ C) of bandwidth, and has preemptive priority over class 2 users. Hence, if n1 class 1 users are in the system, they use an amount n1B1 of the total capacity C. Class 2 users share bandwidth not used by class 1. Hence, if n2 class 2 users are in the system, each of them receives an amount (C ⫺ n1B1)/n2 of the available bandwidth. In the sequel, we assume, with no loss of generality, B1 1. Hence, the service completion rates for the two classes are n1m 1 and (C ⫺ n1)m 2 respectively. Both class 1 and 2 users may be rejected on their arrival. A class 1 user is rejected if the required bandwidth is not available. A class 2 user is rejected if the number of class 2 users already in the system is equal to a given threshold N2. In this paper we do not consider the case of queuing. Hence, a rejected user is lost. As explained in Section 1, it is important to manage the available bandwidth so as to maintain the blocking probability for both classes within acceptable limits. From the point of view of class 1, the best policy would of course be to make available to this class all the capacity C to satisfy arriving requests (note that, because of the preemptive priority, class 1 ‘‘does not see’’ class 2). But this policy could cause unacceptably high blocking probability for class 2.
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Hence, it is opportune to investigate access control policies that limit the access of class 1 connections, by limiting the total bandwidth amount that can be assigned to class 1. Two possible policies (already proposed in the literature in analogous contexts [13, 7, 8, 14]) are based on the fixed threshold and hysteresis concepts. In the fixed threshold policy, class 1 connections can use up to C 0 ⬍ C bandwidth with preemptive priority over class 2, while the remaining C ⫺ C 0 bandwidth is strictly reserved for class 2. The hysteresis policy has been proposed to get more flexibility with respect to a fixed threshold policy, by differentiating the policy behavior under normal load and overload system conditions. In our case, it may be defined by using two thresholds C 0 and C 00 , with 0 ⬍ C 0 ⬍ C 00 ⱕ C. The system starts in normal load condition and until the bandwidth used by class 1 is less than C 00 it remains in that condition. As soon as the bandwidth used by class 1 becomes equal to C 00 the system enters the overload condition, where only class 2 requests are served. The system returns in normal load condition only when the bandwidth used by class 1 drops below C 0 . Note that it may be C 00 C. Hence this policy can make available all the bandwidth to class 1 in normal load condition. Only if class 1 becomes too ‘‘greedy’’ is the system temporarily shut down for class 1, to give room to class 2. The effectiveness of these or other policies must be evaluated with respect to a given objective, defined in terms of the blocking probabilities for the two classes of users. Let PB1 and PB2 denote the blocking probabilities for class 1 and 2 respectively. In this paper we analyze the problem of determining optimal policies for the following objectives: (a) minimizing H·PB1 ⫹ L·PB2, with L, H ⱖ 0; (b1) minimizing PB1, subject to PB2 ⱕ P 00 ; (b2) minimizing PB2, subject to PB1 ⱕ P 0 . To this purpose, we define in the next sections two quite general classes of policies, and provide results concerning the above objectives.
3. Minimizing a linear objective function 3.1. Generalized hysteresis policies Let us consider the two policies described in Section 2. The fixed threshold policy bases its acceptance or rejection decision on the number of present class 1 users. On the other hand, the hysteresis policy bases its decision on an additional state information (normal load or overload), besides the number of present users. We consider a class of access control policies that can be considered as a generalization of these policies, called generalized hysteresis policies (GHP), where each arriving user is accepted or rejected according to the following state-dependent rules.
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(
d i; j; s2
Fig. 1. State transitions for process Xd .
The set of possible system states is defined as E {i; j; s 兩 0 ⱕ i ⱕ C; 0 ⱕ j ⱕ N2 ; s 僆 {1; 2; …; S}}
a 2; s2 ; s2
if i ⬎ C 0
a 2; s2 ; s1
if i ⱕ C 0
(where we assume that label s1 denotes normal load state and label s2 denotes overload state). Both fixed threshold and hysteresis policies choose deterministically a single decision in each state on class 1 arrival or departure. In general, a policy d 僆 GHP class can choose probabilistically among different decisions. We denote by pd i; j; s; a x; s 0 ; s 00 the probability of decision a x; s 0 ; s 00 in state i; j; s under policy d . Of course, it must hold 2 S S X X X
pd i; j; s; a x; s 0 ; s 00 1
In each state [i, j, s], i represents the number of class 1 users, j the number of class 2 users, and s represents a state label. The inclusion of a state label in the state definition allows us to define policies that take different decisions under the same occupancy level i,j. The decision to be taken in each state [i, j, s], with i ⬍ C is basically whether or not to accept an incoming class 1 user. More precisely, the set A of possible decisions in each state [i, j, s] has cardinality 2S 2 and its elements are defined as follows:
Under a given policy d 僆 GHP class, the system state evolution is described by a continuous time Markov process Xd fXd t 兩 ⱖ 0g, with state space E. Fig. 1 shows the possible transitions from a generic state i; j; s 僆 E, where ( l1 pd i; j; s; a x; s 0 ; s 00 ifx 1 0 l1 x; s 0 ifx 2
a 1; s 0 ; s 00 ; s 0 ; s 00 僆 {1; 2; …; S} :
m1 x; s 00 im1 pd i; j; s; a x; s 0 ; s 00
an arriving class 1 user is accepted and causes a transition to state i ⫹ 1; j; s 0 ; whereas a class 1 departure causes a transition to state i ⫺ 1; j; s 00 ; a 2; s 0 ; s 00 ; s 0 ; s 00 僆 {1; 2; …; S} : an arriving class 1 user is rejected and causes a transition to state i; j; s 0 ; whereas a class 1 departure causes a transition to state i ⫺ 1; j; s 00 : Of course, in each state C; j; s only decisions a 2; s 0 ; s 00 are possible. Each policy d 僆 GHP class corresponds to a given choice of the decision to be taken in each state, i.e. it can be defined as a function d : E ! A. In particular, it is easy to realize that the fixed threshold and hysteresis policies are special instances of the GHP class, and correspond to the following decisions choices:
x1 s 0 1 s 00 1
m2 i C ⫺ im2 As it can be noted, the rates for class 1 are policy dependent, as stated. On the other hand, the departure rate for class 2 is defined according to the ‘‘sharing available bandwidth’’ rule, introduced in Section 2. Let us denote by pd i; j; s the steady-state probabilities of Xd . We have the following expressions for the blocking probabilities of classes 1 and 2, denoted by PB1 and PB2 respectively: X X pd e; a·pd e 1 PB1 ei;j;s aa 2;·;·
PB2
X
pd e
2
ei;j;s jN2
• fixed threshold policy: (
d i; j; s
a 1; s; s if i ⬍ C 0 a 2; s; s if i ⱖ C 0
• hysteresis policy: (
d i; j; s1
a 1; s1 ; s1 if i ⬍ C 00 ⫺ 1 a 1; s2 ; s1 if i ⱖ C 00 ⫺ 1
3.2. Optimal policy The definition of the GHP class allows us to formulate under a Markov decision theory framework the problem of determining within this class an optimal policy for the objective (a) introduced in Section 2, i.e. (a) minimizing H·PB1 ⫹ L·PB2, with L, H ⱖ 0. To this purpose, we introduce a cost function r : E × A ! R, where R is the set of real numbers, that
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defines the cost per unit time accumulated by process Xd : 8 0 if x 1 and j ⬍ N2 > > > > >
By using Eqs. (1) and (2) we can rewrite Eq. (4) as follows: X X L·pd e; a·pd e z d ei;j;s
aa 1;·;·
j N2 X ⫹ ei;j;s
⫹
X
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are fixed by the characteristics of the system. Hence, to possibly limit the cardinality of E and consequently the cost of finding the optimal policy, we can only act on parameter S, that gives the number of possible state labels used to define a control policy (e.g. we need S 1 label for the fixed threshold policy, and S 2 labels for the hysteresis policy). Small S values reduce computational costs, but could exclude potentially better policies; the opposite is true with high S values. In the following, we prove that the search for the optimal solution of problem (a) can be limited to the smallest possible subset of DGHP, i.e. the subset obtained by fixing S 1, thus considerably reducing the computational cost. To this purpose, we adopt a technique presented in Ref. [11]. Basically, the steps of this technique are as follows (we refer to Ref. [11] for mathematical details). Instead of considering the infinite horizon non-discounted problem given by Eq. (6), we will consider the corresponding finite horizon discounted problem, will prove some property of the solution of this problem, and will infer equivalent property for the solution of Eq. (6), thanks to relations between the two kinds of problem. Let us denote by ZTa i; j; s the average discounted cost accumulated in [0, T] by process Xd with cost rate function r(·), starting from state i; j; s 僆 E, i.e. ZTa i; j; s ZT
L ⫹ H·pd e; a·pd e
E
aa 2;·;·
0
e⫺at r Xd t; d Xd t dt 兩 Xd 0 i; j; s ;
a⬎0
7
j N2 X
X
ei;j;s
aa 2;·;·
H·pd e; a·pd e
j ⬍ N2 H·PB1 ⫹ L·PB2
5
Let us denote by d * the optimal policy such that z dⴱ min {z d 兩 d 僆 GHP}
6
From Eq. (5) it follows that d * represents the solution for problem (a). Hence, standard algorithms developed in the framework of Markov decision theory can be used to determine this solution [15, 16]. Actually, from theory, we know that d * belongs to the subset DGHP 傺 GHP consisting of deterministic policies only, that choose deterministically a single decision in each state (i.e. for each state i; j; s there exists a single action a x; s 0 ; s 00 such that pd i; j; s; a x; s 0 ; s 00 1). Note that both the fixed threshold and the hysteresis policies defined in Section 2 belong to this subset, and hence are candidates for being the optimal solution. However, it should be noted that the cost of any algorithm used to determine d * is roughly proportional to 兩E兩 3, where: 兩E兩 S·C·N2 [15]. In the above expression for the cardinality of E, C and N2
Hence, the finite horizon discounted problem consists in determining the policy in DGHP that minimizes Eq. (7). Now, we can discretize this problem by using the standard uniformization technique [15]. It is easy to realize that the uniformization rate can be defined as L l1 ⫹ l2 ⫹ C m1 (under the assumption m 1 ⱖ m 2; the complementary assumption m 1 ⬍ m 2 gives rise to the same final results, and will not be considered in the sequel). Let Z ⴱa k; i; j; s be the optimal cost for the k-step discretized problem, starting from state i; j; s 僆 E. Normalizing the transition rates with respect to L , the following equation holds: Z ⴱa k; i; j; s l1 min{{Z ⴱa k ⫺ 1; i ⫹ 1; j; s 0 兩 1 ⱕ s 0 ⱕ S} 傼 {H ⫹ Z ⴱa k ⫺ 1; i; j; s 0 兩 1 ⱕ s 0 ⱕ S}} ⫹ im1 min{Z ⴱa k ⫺ 1; i ⫺ 1; j; s 0 ( ⴱa Z k ⫺ 1; i; j ⫹ 1; s if j ⬍ N2 0 兩 1 ⱕ s ⱕ S} ⫹ l2 L ⫹ Z ⴱa k ⫺ 1; i; j; s if j N2 ⫹ I j C ⫺ im2 Z ⴱa k ⫺ 1; i; j ⫺ 1; s ⫹ C ⫺ i m1 ⫺ I jm2 Z ⴱa k ⫺ 1; i; j; s
8
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where I j
1 if j ⬎ 0 0
if j 0
The first term in Eq. (8) is the contribution to the cost of class 1 arrivals, where we have the option of accepting the user and changing the state label to some s 0 , or rejecting the user, paying a cost H and changing state label to some s 0 ; the second term is the contribution of a class 1 departure, where we have the option of changing the state label to some s 0 ; the third term is the contribution of class 2 arrivals, where a cost L is accumulated only in saturated states; the fourth term is the contribution of a class 2 departure if j ⬎ 0, and the indicator function I( j) takes into account the case j 0. Finally, the last term is a consequence of uniformization. Boundary conditions for class 1 arrivals and departures can be simply handled by assuming Z ⴱa k; C ⫹ 1; j; s ∞. The following theorem holds for the optimal cost function defined by Eq. (8).
from theory that the optimal policy for problem (a) is deterministic, i.e. it chooses a single action in each state with probability 1. In this section, we consider problems (b1) and (b2) of Section 2, defined as (b1) minimizing PB1, subject to PB2 ⱕ P 00 ; (b2) minimizing PB2, subject to PB1 ⱕ P 0 . For these problems we consider a class of policies that base their decisions only on the occupancy levels for the two classes and that also includes probabilistic policies. We show that the optimal policies are probabilistic and also give algorithms to determine these policies. 4.1. Generalized fixed threshold policies The policies we consider consists of threshold-based policies that use a fractional threshold value to determine the acceptance or rejection of a class 1 user. More formally, these state-dependent policies can be defined as follows. The system state space is defined as E {i; j 兩 0 ⱕ i ⱕ C; 0 ⱕ j ⱕ N2 }
ⴱa
0
ⴱa
00
Theorem 1. Z k; i; j; s Z k; i; j; s for any s 0 ; s 00 僆 {1; 2; …; S}
Proof. We prove the theorem by induction on k. The basic step is trivial since Z ⴱa 0; i; j; s 0 0 for any s. Let us assume the theorem is true for Z ⴱa k ⫺ 1; ·. By using this induction hypothesis, it is straightforward to see from Eq. (8) that the theorem also holds for Z ⴱa k; ·. QED According to the results reported in Ref. [11], the property proved in Theorem 1 also applies to zⴱ z dⴱ defined by Eq. (6), since the state space of Xd is finite and Xd is irreducible. Hence, Theorem 1 states that the best policy in DGHP (such that Xd is irreducible) that uses state labels does not behave better (with respect to objective (a) of the best policy that bases its decision only on the occupancy level. As a consequence, we can limit the search for the optimal policy for problem (a) to the subset of DGHP consisting of policies that use no state label in their state description, thus reducing by a factor S the cardinality of the state space and by a factor S 2 the cardinality of the set of actions. Finally, note that, with respect to the two policies described in Section 2, this means that the hysteresis policy (that needs two different state labels for its implementation) cannot do better than single label policies (i.e. with S 1). 4. Minimizing blocking probability with constraint In Section 3 we have shown that the optimal policy for problem (a) takes its decisions on the basis only of the occupancy levels for the two classes. Moreover, we know
In each state i; j, i represents the number of class 1 users and j the number of class 2 users. The decision to be taken in each state i; j, with i ⬍ C is whether to accept or not an incoming class 1 user. Hence the set A of possible decisions in each state i; j is: • a(1): an arriving class 1 user is accepted and causes a transition to state i ⫹ 1; j; • a(2): an arriving class 1 user is rejected and causes no state transition. Of course, in each state C; j only decision a(2) is possible. The class of policies we consider, called generalized fixed threshold policies (GFTP), consists of policies d dn;p , defined as follows: 8 a 1 if i ⬍ n > > > > > < a 1 with probability p; if i n dn;p i ; j > > a 2 with probability 1 ⫺ p; if i n > > > : a 2 if i ⬎ n As it can be noted, each policy d n,p 僆 GFTP can be considered as a fixed threshold policy with fractional threshold t, where the integer part of t is equal to n and the fractional part is equal to p. Let Xd;n;p be the continuous time Markov process that describes the system behavior under policy d n,p. In Fig. 2 we show the possible transitions from a generic state i; j, where 8 l1 if i ⬍ n > > < l1 i pl1 if i n > > : 0 otherwise
m2 i C ⫺ im2
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Hence, the theorem can be proved by proving separately that PB1(n, p) is a decreasing function of n, for p 0, and PB1(n, p) is a decreasing function of p, for a given n. Let us prove that PB1 n; 0 ⬎ PB1 n ⫹ 1; 0. By using Eq. (9), and by some simple algebra, this is equivalent to proving that nX ⫹1 j0
which is true, since it holds that
Fig. 2. State transitions for process Xd ,n,p.
Now, we show that the optimal policies for objectives (b1) and (b2) do belong properly to GFTP, in the sense that an integer threshold n does not suffice, in general, to solve these problems. To this purpose, in the following we first prove that PB1 is a decreasing function of t, whereas PB2 is an increasing function of t. Then, we use these results to argue that the optimal thresholds for problems (b1) and (b2) are fractional, and define simple algorithms to determine these thresholds. Theorem 2. PB1 is a decreasing function of t n ⫹ p, n ⬍ C, 0 ⱕ p ⱕ 1. Proof. Let us consider the policy d n,p. Remember that, because of the preemptive priority, class 1 users do not see class 2 users. Hence, the class 1 behavior under this policy is described by the birth-and-death process whose state diagram is shown in Fig. 3. Let us denote by p (i) the steady state probability of state i for this process. We have [17]
p i
ri p 0; i!
p n ⫹ 1 p 2
p 0 4
0ⱕiⱕn
rn⫹1 p 0 n ⫹ 1!
nX ⫹1 j0
nX ⫹1 n X rj rj rj 1⫹ 1⫹r j! j! j ⫹ 1! j1 j0
⬎1⫹
n r X rj n ⫹ 1 j0 j!
Now, we show that PB1 n; p is decreasing with respect to p. Differentiating Eq. (9) with respect to p we have " # 2 rn rn⫹1 rn⫹1 ⫹p PB1 n; p ⫺ 1 ⫺ p n! n ⫹ 1! n ⫹ 1! 2p 2
3⫺2 " # j n⫹1 n⫹1 n n X r r r r 5 ⫹ ⫹p ⫺ 4 j! n ⫹ 1! n ⫹ 1! n! j0 2
3⫺1 j n⫹1 n X r r 5 ⫹p 4 j! n ⫹ 1! j0 By some simple algebra, it is easy to see that the condition (2/2p)PB1(n, p) ⬍ 0 is equivalent to the condition n nX ⫹1 r X rj rj ⬍ n ⫹ 1 j0 j! j! j0
which is true, as proved above. QED
3⫺1
n X rj rn⫹1 5 ⫹p j! n ⫹ 1! j0
To prove that PB2 is an increasing function of t is somewhat more complicated, since class 2 users ‘‘see’’ class 1 users. We proceed as follows. Let us define the following cost function for the process Xd ,n,p shown in Fig. 2: ( 0 if j ⬍ N2 r i; j 1 if j N2
where
r
n rj r X rj ⬎ j! n ⫹ 1 j0 j!
l1 m1
Then, it is PB1 n; p 1 ⫺ pp n ⫹ p n ⫹ 1
9
Note that PB1(n, p) is a continuous function of p and n.
Fig. 3. Behavior of class 1 under policy d n,p.
Let zd ,n,p be the average cost per unit time for Xd ,n,p, defined as ZT E r Xd;n;p t dt 0 zd;n;p lim T!∞ T It is easy to see that PB2 n; p zd;n;p . Now, we can transform by uniformization Xd ,n,p into an equivalent discrete time process, with uniformization rate L l1 ⫹ l2 ⫹ C m1 (as
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in Section 3, we assume m 1 ⱖ m 2). Without loss of generality, let us assume L 1. Let zd;n;p k; i; j denote the average cost accumulated in k steps by the uniformized process obtained from Xd ,n,p, starting from state i; j. The following equation holds: zd;n;p k; i; j r i; j ⫹ l1 8 zd;n;p k ⫺ 1; i ⫹ 1; j > > < × pzd;n;p k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j > > : zd;n;p k ⫺ 1; i; j
( ⫹ l2 ×
zd;n;p k ⫺ 1; i; j ⫹ 1
if j ⬍ N2
zd;n;p k ⫺ 1; i; j
if j N2
if i ⬍ n if i n if i ⬎ n
⫹ I j C ⫺ im2 zd;n;p k ⫺ 1; i; j ⫺ 1 10
where I(j) is defined as in Section 3.2 and takes into account the case in which there are no departures of class 2 because j 0. Moreover, it also holds that zd;n;p
zd;n;p k; i; j lim k!∞ k
11
Now, we need to prove the following two lemmas. Lemma 1. for all k. Proof.
zd;n;0 k; i; j is monotonically increasing in j
See Appendix A.
Lemma 2. for all k.
Proof. We prove the theorem by proving separately that zd;n;0 k; i; j is an increasing function of n, for each k, and that zd;n;p k; i; j is an increasing function of p, for each k. Then, from Eq. (11) and PB2 zd;n;p the theorem is proved. Let us prove by induction that zd;n⫹1;0 k; i; j ⫺ zd;n;0 k; i; j ⱖ 0. The basic step is trivially true since zd;n;0 0; i; j 0. Let us assume the theorem is true for k ⫺ 1. Let us consider zd;n;0 k; i; j. We have zd;n⫹1;0 k; i; j ⫺ zd;n;0 k; i; j l1 8 zd;n⫹1;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i ⫹ 1; j if i ⬍ n > > < if i n × zd;n⫹1;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j > > : zd;n⫹1;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i; j if i ⬎ n
⫹ im1 zd;n;p k ⫺ 1; i ⫺ 1; j
⫹ C ⫺ i m1 ⫺ I jm2 zd;n;p k ⫺ 1; i; j
Theorem 3. PB2 is an increasing function of t n ⫹ p, n ⬍ C, 0 ⱕ p ⱕ 1.
zd;n;0 k; i; j is monotonically increasing in i
Proof. See Appendix A. By using the two lemmas, we can prove the following theorem.
⫹l 2
( ×
zd;n⫹1;0 k ⫺ 1; i; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i; j ⫹ 1
if j ⬍ N2
zd;n⫹1;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i; j
if j N2
⫹im1 zd;n⫹1;0 k ⫺ 1; i ⫺ 1; j ⫺ zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹I j C ⫺ im2 zd;n⫹1;0 k ⫺ 1; i; j ⫺ 1 ⫺zd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ C ⫺ i m1 ⫺ I jm2 zd;n⫹1;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i; j
All the differences in the right-hand side of the above equation are ⱖ 0 by induction hypothesis, with the exception of the second difference in the first term (case i n); for this difference we have zd;n⫹1;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j ⱖ zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j by induction hypothesis ⱖ 0 by Lemma 2
Now, let us prove by induction that zd;n;p k; i; j ⫺ zd;n;q k; i; j ⱖ 0 for 0 ⱕ q ⬍ p ⱕ 1. Again, the basic step is trivially true. Then, we have
zd;n;p k; i; j ⫺ zd;n;q k; i; j 8 if i ⬍ n zd;n;p k ⫺ 1; i ⫹ 1; j ⫺ zd;n;q k ⫺ 1; i ⫹ 1; j > > < l1 × pzd;n;p k ⫺ 1; i ⫹ 1; j ⫺ qzd;n;q k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j ⫺ 1 ⫺ qzd;n;q k ⫺ 1; i; j if i n > > : zd;n;p k ⫺ 1; i; j ⫺ zd;n;q k ⫺ 1; i; j if i ⬎ n ( zd;n;p k ⫺ 1; i; j ⫹ 1 ⫺ zd;n;q k ⫺ 1; i; j ⫹ 1 if j ⬍ N2 ⫹ im1 zd;n;p k ⫺ 1; i ⫺ 1; j ⫺ zd;n;q k ⫺ 1; i ⫺ 1; j ⫹l2 × zd;n;p k ⫺ 1; i; j ⫺ zd;n;q k ⫺ 1; i; j if j N2 ⫹I j C ⫺ im2 zd;n;p k ⫺ 1; i; j ⫺ 1 ⫺ zd;n;q k ⫺ 1; i; j ⫺ 1
⫹ C ⫺ i m1 ⫺ I jm2 zd;n;p k ⫺ 1; i; j ⫺ zd;n;q k ⫺ 1; i; j
V. de Nitto Persone`, V. Grassi / Computer Communications 21 (1998) 1559–1570
Fig. 4. Optimal threshold for problem (b1).
All the differences in the right-hand side of the above equation are ⱖ 0 by induction hypothesis, with the exception of the second difference in the first term (case i n); for this difference we have
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else begin t_inf: 0; t_sup: C; repeat t: (t_sup ⫹ t_inf)/2; if PBx(t) ⬍ P then begin t_inf: t; diff: t_sup ⫺ t end else begin t_sup: t; diff: t ⫺ t_inf end until diff ⱕ 1; return(t) end; end.
pzd;n;p k ⫺ 1; i ⫹ 1; j ⫺ qzd;n;q k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j ⫺ 1 ⫺ qzd;n;q k ⫺ 1; i; j ⱖ pzd;n;p k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ pzd;n;p k ⫺ 1; i; j ⫺ qzd;n;p k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ qzd;n;p k ⫺ 1; i; j by induction hypothesis p ⫺ q zd;n;p k ⫺ 1; i ⫹ 1; j ⫺ zd;n;p k ⫺ 1; i; j ⱖ 0 by Lemma 2 and p ⬎ q
(Note that, actually, Lemma 2 has been proven for the special case p 0; the extension to the case p ⬎ 0 is trivial and is omitted.) QED Now, let us consider problem (b1): minimizing PB1, subject to PB2 ⱕ P 00 . From Theorems 2 and 3 we have that the behavior of PB1 and PB2 as a function of t is as depicted in Fig. 4. Hence, any value t ⱕ t00 (shown by the shadowed region) satisfies the constraint PB2 t ⱕ P 00 . Then, the optimal value for t is t t00 , since PB1 t is a decreasing function of t. This means that the optimal policy in the class GFTP for problem (b1) is the policy d1 dn 00 ;p 00 · such that PB2 n 00 ⫹ p 00 P 00
12
Analogously, the optimal policy in the class GFTP for problem (b2) (minimizing PB2, subject to PB1 ⱕ P 0 ), is the policy d2 dn 0 ;p 0 · such that: PB1 n 0 ⫹ p 0 P 0
13
It is easy to be convinced that the solution of Eqs. (12) and (13) is, in general, non-integer, thus proving our statement that the optimal threshold for problems (b1) and (b2) is fractional. In the following, we give a simple algorithm to determine the optimal thresholds t00 and t0 , based on a binary search of the solutions of Eqs. (12) and (13) respectively. The algorithm is guaranteed to converge to the solutions thanks to the monotonicity properties of PB2 and PB1. Algorithm B x; P; b begin if PBx(b) ⬎ P then no solution
The algorithm parameters are (x 2, P P 00 , b 0), and (x 1, P P 0 , b C) for problems (b1) and (b2) respectively, whereas 1 is the required resolution for the problem solution (e.g. 1 10 ⫺4). Note that in algorithm B 1; P 0 ; C, PB1(t) can be evaluated by using the closedform expression (9) in Theorem 2. On the other hand, no closed-form expression is available for the evaluation of PB2(t) in algorithm B 2; P 00 ; 0; we have to solve the balance equations of Markov process Xd;n;p (with n ⫹ p t) to calculate PB2(t). Hence, each step in the iterative part of algorithm B 2; P 00 ; 0 has a greater complexity than the corresponding step of algorithm B 1; P 0 ; C (in other words, solving problem (b1) costs more than solving problem (b2)). 4.2. Numerical results In Tables 1–4 we present some numerical results obtained by algorithm B x; P; b. Tables 1 and 2 refer to problem (b1), with constraint PB2 ⱕ 0.05; Tables 3 and 4 refer to problem (b2), with constraint PB1 ⱕ 0.01. In all the tables, column t_opt reports the optimal threshold calculated by the algorithm and columns PB1(t_opt) and PB2(t_opt) the corresponding blocking probabilities. Note that, by the algorithm definition, the value t_opt is such that the constraint value is exactly met by the corresponding blocking probability; hence, we have PB2(t_opt) 0.05 in Tables 1 and 2, and PB1(t_opt) 0.01 in Tables 3 and 4. However, the actual implementation of a fractional threshold may be somewhat more complex than the implementation of an integer threshold. As a consequence, an alternative choice (Choice 1) could be to use as threshold the closest integer that satisfies the constraint. In this case, it
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Table 1 Problem (b1): C 20, N2 20, B1 1, l 1 10, m 1 1, m 2 1 PB2 ⱕ 0.05
l2 10 11 12 14
t_opt
14.872 12.067 10.141 7.122
PB1(t_opt)
0.0390 0.1172 0.2062 0.3968
PB2(t_opt)
0.05 0.05 0.05 0.05
Choice 1 cost %PB1(t_inf)
Choice 2 cost %PB2(t_closest)
⫹ 45 ⫹2 ⫹4 ⫹3
⫹3 — — —
is useful to evaluate the ‘‘cost’’ of such a choice, which may be measured as the corresponding percentage increase of the non-constrained blocking probability. The fifth columns of Tables 1–4 (labeled ‘‘Choice 1 cost’’) shows this percentage increase when using as threshold the closest integer that satisfies the constraint. This integer is equal to bt_optc (t_inf) for Tables 1 and 2, whereas it is equal to dt_opte (t_sup) for Tables 3 and 4. As it can be noted, the percentage increase ranges from 0.2 to 45%, with values below 10% in most of the cases. These results suggest that in many cases this integer threshold may be satisfactory. However, it is worth pointing out that Choice 1, i.e. the use of the closest integer that satisfies the constraint, can yield a threshold that is quite far from the optimal one. This can be noted, for example, from the first row of Table 1 (where t_opt 14.872 and t_inf 14), or from Table 4 (where t_opt 16.2 and t_sup 17). It can also be noted that, in some of these cases, the cost of Choice 1 is quite high, as shown by the fifth column values. Thus, if an integer threshold is considered preferable for implementation simplicity, this suggests as alternative choice the use of the integer closest to the optimal threshold (Choice 2), even if this can lead to violating the given constraint. In this case, the ‘‘cost’’ of Choice 2 may be evaluated by measuring how much the constraint is violated. Hence, the last columns of the tables (labeled Choice 2 cost) show the impact of using the integer closest to the optimal fraction (t_closest). In all the tables, this last columns show the percentage increase of the constrained probability PBx with respect to the constraint value (0.05 for Tables 1 and 2, and 0.01 for Tables 3 and 4), only in the cases where the use of t_closest leads to a constraint violation (in the other cases, Choice 2 coincides with Choice 1). In the case of the first row of Table 1, we see that the cost of Choice 2 (value of %PB2(t_closest)) results equal to 3%.
Table 3 Problem (b2): C 20, N2 20, B1 1, l 2 10, m 1 1, m 2 1, PB1 ⱕ 0.01
l1
t_opt
PB1 (t_opt)
PB2(t_opt)
Choice 1 cost %PB2(t_sup)
Choice 2 cost %PB1(t_closest)
8 9 10 11 12
14.889 16.200 17.506 18.756 19.969
0.01 0.01 0.01 0.01 0.01
0.0118 0.0311 0.0689 0.1283 0.2055
⫹ 1.7 ⫹ 6.8 ⫹ 3.2 ⫹ 1.2 ⫹ 0.2
— ⫹ 10 — — —
This small cost may be weighed against the cost of Choice 1 (value of %PB1(t_inf)), that results equal to 45%. On the other hand, we see in Table 4 that the Choice 2 cost results equal to 10%, against a cost of Choice 1 (value of %PB2(t_sup)) that ranges from 3 to 28%. Finally, it is worth noting that when either Choice 1 or 2 is adopted, Algorithm B x; P; b is still useful to determine the corresponding integer threshold.
5. Conclusions We have analyzed the problem of access control in a wireless cellular environment that supports multiple classes of traffic with different QoS requirements. The class with more critical requirements needs to be privileged in resource assignment; however, to avoid abuse of this privilege it is opportune to implement some form of control on the access requests of the privileged class. Hence, we have studied the problem of determining optimal access control policies with respect to two different objectives, both based on the blocking probabilities for the considered classes of users. The blocking probability is an important metric to measure the effectiveness of control policies in a microcellular environment, where a large fraction of the incoming traffic consists of handoff requests of already active connections, whose interruptions should be avoided as much as possible, to avoid users dissatisfaction. For the first objective (minimizing a linear function of the blocking probabilities), we have defined a very general class of control policies, and we have shown that the optimal policy belongs to a small subset of this class, thus improving the efficiency of standard algorithms used to determine this policy. For the Table 4 Problem (b2): C 20, N2 20, B1 1, l 1 9, m 1 1, m 2 1, PB1 ⱕ 0.01
Table 2 Problem (b1): C 20, N2 20, B1 1, l 2 10, m 1 1, m 2 1, PB2 ⱕ 0.05
l1
t_opt
PB1(t_opt)
PB2(t_opt)
Choice 1 cost %PB1(t_inf)
11 12 13
13.239 12.463 12.033
0.1100 0.1765 0.2355
0.05 0.05 0.05
⫹7 ⫹ 12.5 ⫹ 0.8
Choice 2 cost %PB2(t_closest) — — —
l2
t_opt
PB2(t_opt)
PB2(t_opt)
Choice 1 cost %PB2(t_sup)
Choice 2 cost %PB1(t_closest)
7 8 9 10 11 12
16.200 16.200 16.200 16.200 16.200 16.200
0.01 0.01 0.01 0.01 0.01 0.01
0.000604 0.00317 0.0115 0.0311 0.0650 0.111
⫹ 28 ⫹ 18 ⫹ 11 ⫹ 6.8 ⫹ 3.4 ⫹3
⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10 ⫹ 10
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second objective (minimizing the blocking probability of a class, with a constraint on the blocking probability of the other class), we have proved results that have allowed us to define a simple algorithm that determines the optimal threshold for admission control. Since the resulting optimal threshold is fractional, we have also analyzed the impact of approximating this threshold by an integer threshold, that is simpler to implement.
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Acknowledgements This work has been partially supported by the CNR project ‘‘Ambienti avanzati per comunicazioni a pacchetto’’, and by MURST funds. We would like to thank the anonymous referees for their suggestions that helped us to improve the paper.
Appendix A
Lemma 1.
zd;n;0 k; i; j is monotonically increasing in j for all k.
Proof. We have to prove that zd;n;0 k; i; j ⫹ 1 ⱖ zd;n;0 k; i; j. We prove the lemma by induction on k. The basic step is trivial since zd;n;0 0; i; j 0 for any [i, j]. Let us assume the lemma is true for k ⫺ 1. We have zd;n;0 k; i; j ⫹ 1 ⫺ zd;n;0 k; i; j r i; j ⫹ 1 ⫺ r i; j ⫹ l1 ( zd;n;0 k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i ⫹ 1; j if i ⬍ n × if i ⱖ n zd;n;0 k ⫺ 1; i; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i; j ( zd;n;0 k ⫺ 1; i; j ⫹ 2 ⫺ zd;n;0 k ⫺ 1; i; j ⫹ 1 if j ⬍ N2 ⫺ 1 ⫹ l2 × 0 if j N2 ⫺ 1 ⫹ im1 zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹ C ⫺ im2 zd;n;0 k ⫺ 1; i; j ⫺ I jzd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ C ⫺ i m1 ⫺ m2 zd;n;0 k ⫺ 1; i; j ⫹ 1 ⫺ C ⫺ i m1 ⫺ I jm2 zd;n;0 k ⫺ 1; i; j ⱖ 0 since, by induction hypothesis, all the differences in the right-hand side of the above equation are ⱖ 0. QED
Lemma 2.
zd;n;0 k; i; j is monotonically increasing in i for all k.
Proof. We have to prove that zd;n;0 k; i ⫹ 1; j ⱖ zd;n;0 k; i; j. We prove the lemma by induction on k. The basic step is trivial since zd;n;0 0; i; j 0 for any [i, j]. Let us assume the lemma is true for k ⫺ 1. We have zd;n;0 k; i ⫹ 1; j ⫺ zd;n;0 k; i; j r i; j ⫹ 1 ⫺ r i; j ⫹ l1 8 zd;n;0 k ⫺ 1; i ⫹ 2; j ⫺ zd;n;0 k ⫺ 1; i ⫹ 1; j if i ⬍ n ⫺ 1 > > < if i n ⫺ 1 × 0 > > : if i ⬎ n ⫺ 1 zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j ( zd;n;0 k ⫺ 1; i ⫹ 1; j ⫹ 1 ⫺ zd;n;0 k ⫺ 1; i; j ⫹ 1 if j ⬍ N2 ⫹ l2 × zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j if j N2 ⫹ m1 i ⫹ 1zd;n;0 k ⫺ 1; i; j ⫺ izd;n;0 k ⫺ 1; i ⫺ 1; j ⫹ I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ 1 ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ m1 ⫺ I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j The first three terms in the right-hand side of the above equation are ⱖ 0 by definition of r(·) and by induction hypothesis. Let
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us consider the remaining three terms; we can rearrange them as follows:
m1 i ⫹ 1zd;n;0 k ⫺ 1; i; j ⫺ izd;n;0 k ⫺ 1; i ⫺ 1; j ⫹I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ 1 ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ m1 ⫺ I jm2 C ⫺ i ⫺ 1zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ C ⫺ izd;n;0 k ⫺ 1; i; j im1 zd;n;0 k ⫺ 1; i; j ⫺ zd;n;0 k ⫺ 1; i ⫺ 1; j ⫹I j C ⫺ i ⫺ 1m2 zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ 1 ⫺ zd;n;0 k ⫺ 1; i; j ⫺ 1 ⫹ m1 ⫺ I jm2 C ⫺ i ⫺ 1 zd;n;0 k ⫺ 1; i ⫹ 1; j ⫺ zd;n;0 k ⫺ 1; i; j ⫹m1 zd;n;0 k ⫺ 1; i; j ⫺ I jm2 zd;n;0 k ⫺ 1; i; j ⫺ 1 ⫺ m1 ⫺ I jm2 zd;n;0 k ⫺ 1; i; j ⱖ 0 by induction hypothesis and by Lemma 1. QED References [1] P. Agrawal, et al., SWAN: a mobile multimedia wireless network, IEEE Personal Communications 3 (2) (1997) 18–33. [2] V.O.K. Li, X. Qiu, Personal communication systems (PCS), Proceedings of the IEEE 83 (9) (1995) 1210–1243. [3] A. Allwan, et al., Adaptive mobile multimedia networks, IEEE Personal Communications 3 (2) (1997) 34–51. [4] D. Towsley, Providing quality of service in packet switched networks, in: L. Donatiello, R. Nelson (Eds.), Performance Evaluation of Computer and Communication Systems, Springer, 1993, pp. 560– 586. [5] W.C.Y. Lee, Smaller cells for greater performance, IEEE Communication Magazine, November, 1991. [6] A. Iwata, N. Mori, C. Ikeda, H. Suzuki, M. Ott, ATM connection and traffic management schemes for multimedia internetworking, Communications of the ACM 38 (2) (1995) 72–89. [7] D. McMillan, Delay analysis of a cellular mobile priority queuing system, IEEE/ACM Trans. on Networking 3 (3) (1995) 310–319. [8] R.G. Schehrer, On a cut-off priority queuing system with hysteresis and unlimited waiting room, Computer Networks and ISDN Systems 20 (1990) 45–56. [9] M. Naghshineh, A.S. Acampora, QoS provisioning in micro-cellular networks supporting multiple classes of traffic, Wireless Networks 2 (3) (1996) 195–204. [10] J.M. Capone, I. Stavrakakis, Achievable QoS and scheduling policies for integrated services wireless networks, Performance Evaluation 2788 (1996) 347–365. [11] R. Ramjee, D. Towsley, R. Nagarajan, On optimal call admission control in cellular networks, Wireless Networks 3 (1) (1997) 29–42. [12] D.E. Comer, D.L. Stevens, Internetworking with TCP/IP, Prentice Hall, 1984. [13] R. Guerin, Queuing-blocking system with two arrival streams and guard channels, IEEE Trans. on Communications 36 (2) (1988) 153–163.
[14] Q.-A. Zeng, K. Mukumoto, A. Fukuda, Performance analysis of mobile cellular radio system with priority reservation handoff procedures, Proc. of IEEE VTC 94 Conference, 1994, pp. 1829–1833. [15] D.P. Heyman, M.J. Sobel, Stochastic Models in Operations Research, McGraw-Hill, 1984. [16] S. Ross, Applied Probability Models with Optimization Applications, Holden-Day, 1970. [17] L. Kleinrock, Queuing Systems, vol. I: Theory, Wiley, 1976.
Vittoria de Nitto Persone` received the Laurea degree in Computer Science from the University of Pisa, Italy, in 1984. She is currently Assistant Professor at the Department of Computer Engineering, University of Roma ‘‘Tor Vergata’’, Roma, Italy. Her main research interests include performance modeling and evaluation, computer and communication systems, wireless systems, distributed systems and parallel processing. She has published several journal and conference papers and participated in the organizing and scientific committees of various international conferences. E-mail: [email protected], [email protected]
Vincenzo Grassi received the Laurea degree in Computer Science from the University of Pisa, Italy, in 1984. He is currently Associate Professor at the Department of Computer Engineering, University of Roma ‘‘Tor Vergata’’, Italy. His research interests include performance and reliability analysis of computer and communication systems, quality of service analysis in high speed and wireless networks, design of distributed and nomadic applications. Professor Grassi is a member of ACM. E-mail: [email protected], [email protected]