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Performance analysis of a mobile communication network: the tandem case
Performance analysis of a mobile communication network: the tandem case Attahiru Sule Alfaa,*, Bin Liub a
Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 b Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, P.R. China Received 9 August 2002; revised 10 June 2003; accepted 10 June 2003
Abstract This paper investigates mobile communication networks consisting of N-cells in tandem. Two models with fixed channel assignment are considered, which correspond to (A) a uni-directional traffic flow and (B) bi-directional traffic flows, respectively. We develop an approximate method to obtain main performance measures of systems such as the loss probabilities of handover calls and fresh calls, and the expected number of occupied channels in each cell. Our approximation is based on decomposing the N-cells in tandem network into N 2 1 pairs of cells with overlaps. The stochastic correlation among neighboring pairs are captured by appropriately selecting the state-dependent Poisson processes as the approximation of handover processes. Some numerical examples are given to demonstrate the accuracy and the convergence of the proposed approach. q 2003 Elsevier B.V. All rights reserved. Keywords: Tandem network; Mobile communication networks; Performance; Corridor movements; Fixed channel allocations; Call loss probabilities
1. Introduction The rapid development of personal communication services (PCS) has stimulated increasing interests in studying the performance measures of cellular mobile communication networks with various control mechanisms. Specifically, this trend has called for better and more efficient analysis of such systems in order to improve the quality of the service. One of the common features in these networks is that the whole service area is divided into cells (Fig. 1), which are served by cell transceivers. However, the number of transceivers (Fig. 2) in each cell is severely limited. This is due to the fact that the number of frequencies available to carry mobile calls is limited. The second feature of these networks is that there are usually two or more classes of calls, e.g. fresh calls and handover calls. The fresh calls are ones which are just starting, and the handover calls are ones which are already ongoing but have moved out of the original cell and need to connect to a transceiver in a new cell. Because of the limited number of channels available for each cell, one of the crucial performance measures for * Corresponding author. Tel.: þ1-204-474-8789; fax: þ 1-204-261-4639. E-mail addresses: [email protected] (A.S. Alfa), [email protected]. ac.cn (B. Liu). 0140-3664/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0140-3664(03)00216-0
cellular mobile communication networks is the loss probability, i.e. the probability that a call cannot be connected. There are two types of common call losses. One is called fresh call loss due to lack of available channels to accommodate a fresh call. The other is called handover call loss. If a mobile terminal moves from the radio coverage of one cell to the radio coverage of another cell, the call is handed over from one transceiver to another. However, if there is no channel available in the new cell, the call is lost. In the perspective of communication company, the latter incurs greater loss than the former. Most models for the analysis of PCS and mobile communication systems have focused only on a single cell. They assume that all the cells in the network are homogeneous (see Refs. [13,14] and the reference therein). Such an assumption is only a matter of convenience for computational tractability, but is not quite realistic. For example, the input of new calls in each cell could be very different and thus immediately contradicts the assumption of homogeneity, especially if the differences are significant. Besides, depending on the designs, the number of channels in each cell are not necessarily the same and the number of channels reserved for handoff calls may also be different. Some researchers recently have considered non-homogeneous cells in the mobile network
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
Fig. 1. A two-directional highway covered by a cellular mobile communication network.
(see Refs. [1,2,4 – 12]). However, most results were obtained only for those networks with product form solutions. Pallant and Taylor [10] studied a cellular mobile network with dynamic channel allocation (maximum packing strategy proposed by Everitt and Macfadyen [4]), and proved that its product form solution exists under certain reversibility conditions. Their model was further extended to allow directed retry for handover calls after being blocked. In Ref. [2], Boucherie and Van Dijk presented a stochastic model for cellular mobile communications networks by taking into account phase-type distributed call-length and phase-type distributed channel holding time. They formulated a stochastically equivalent queueing network with age-dependent routing for their model. For some special networks having product form solutions, they obtained two results on insensitivity of distributions, i.e. the stationary distribution of system depends on the call-length distribution only through its mean, and also depends on the distribution of the time a call spends in cells only through its mean. Boucherie and Mandjes [1] introduced a simulation algorithm to evaluate performance measures (blocking probabilities) from the product form equilibrium distribution. Their numerical method is based on importance sampling in conjunction with large deviation techniques. The necessary condition for using their method is the existence of a product form equilibrium distribution. For some important networks with different protocols such as redial, soft capacity and push-out policy, the authors studied the conditions to ensure the equilibrium distributions to be of the product form.
209
In general, the conditions which ensure a product form equilibrium distribution are quite restrictive and are unlikely to be satisfied in practical. In fact, for the vast majority of networks, the solutions cannot be obtained in closed form. Therefore, it is very important to develop effective approximate methods to obtain their solutions. Massey and Whitt [8] and Leung et al. [7] studied a so-called highway Poisson-arrival-location models (PALM) for a wireless network along a highway, which is closely related to our models. In their model, vehicles were classified as non-calling or calling. Basic model is for traffic on a one-way, semi-infinite highway, with movement specified by a deterministic location function. Two-way model and other more general movements in R2 and R3 were treated as superpositions of independent one-way traffic along paths in these spaces. Under assumption of no capacity constraints, they derived the partial differential equations (PDEs) or the ordinary differential equations (ODEs) to describe the evolution of the system. The call density and call handoff rate were calculated by solving these equations numerically. For models with capacity restrictions, the blocking probabilities of calls were approximated by applying the results on infinite capacity models. No discussion on the accuracy of approximation is provided in their papers. In this paper we consider a network of N-cells in tandem which are prevalent in mobile communication services on highways and railways during the morning and afternoon rush hours. We study two different models with fixed channel allocation, which correspond to (A) a unidirectional mobile traffic flow, and (B) bi-directional mobile traffic flows, respectively. There are two different types of call (handover call and fresh call) arrivals for each cell. To ensure that handover calls gain a higher priority than fresh calls, we always reserve some channels in each cell for possible arriving handover calls (see Refs. [13,14]). An approximation approach is developed to calculate some important performance measures such as the loss probabilities of fresh calls and handover calls and the expected number of occupied channels in each cell. The approximation is based on decomposing the N-cells in tandem to N 2 1 pairs of cells with overlaps. The approximation performs reasonably well in a wide range of data based on the numerical examples. This method has been used
Fig. 2. Communication between mobile PC terminals and transceivers.
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successfully by Brandwajn and Jow [3] to analyze a tandem queueing system with blocking. The rest of the paper is organized as follows. In Section 2, we describe our two models in details. We develop an approximation approach to calculate the stationary distribution of systems and their performance measures in Section 3. In order to test our approximation method, we make some comparisons between approximate results and exact results by aid of numerical examples in Section 4. The paper is concluded in Section 5.
in node j reaches (greater than or equal to) mj ; any arriving fresh calls at node j from outside will be refused access and will be lost naturally. Once a call (fresh or handover) is accepted at node j; it will stay there and occupy a channel for a random time until its service is completed or the mobile vehicle moves out of this cell. This time is referred to as channel holding time at node j: We assume that the channel holding time at node j is exponentially distributed with rate mj : 2.2. Model B: bi-directional mobile traffic flows
2. Description of models 2.1. Model A: a uni-directional mobile traffic flow Consider a corridor of travel which consists of N-cells. This can be transformed into a mobile communication network (queueing system) consisting of N nodes in tandem, where the cells are the nodes. From now on, we use nodes and cells interchangeably. All mobile terminals move in a uni-direction. Without loss of generality, we assume that they move in increasing order of node labels. Node j ðj ¼ 1; …; NÞ has Kj channels and no waiting queues are allowed. Namely, any arriving call, which finds no idle channel available, will be lost automatically. For node j ðj ¼ 2; …; NÞ there are two different types of call arrivals, i.e. handover calls which are handed over from node j 2 1 and fresh calls which come from outside system (or originate inside the jth cell). As a matter of fact, the network we discuss here should be viewed as a part of global network, so there are still these two kinds of call arrivals at node 1. Assume aj to be the arrival rate of fresh calls at node j; and l to be the arrival rate of handover calls at node 1. After the completion of service in node j a call leaves voluntarily from the system with probability pj ; and/or with probability qj ¼ 1 2 pj it goes into node j þ 1 if there is any channel available there, otherwise it will be forced to leave from system. It is worth to note that there are two types of different calls leaving phenomena. One is due to its whole service completion, i.e. this call is fully completed. The other is due to the fullness of the downstream node. For each node j there is a threshold value which is used as a control if an arriving fresh call will be accepted or not. More precisely, as long as the number of occupied channels
There are two-directional traffic flows moving through a N-node tandem network. Namely, some of the mobile terminals move in directions along with the increasing order of node labels (toward right), the others with the decreasing order of node labels (toward left), i.e. in the opposite direction. We name those calls moving towards the right (left) hand side as type-1(2) calls. Assume the service times of type-1 (type-2) calls in node j have exponential distributions with rates m1;j ðm2;j Þ: The arrival rate of type-1 (type-2) fresh calls at node j are aj ðbj Þ: After the completion of service in node j each type-1 (type-2) call leaves voluntarily from the system with probability p1;j ðp2;j Þ; and/or with probability q1;j ðq2;j Þ it goes into node j þ 1 ðj 2 1Þ if there is any channel available there, otherwise it will be forced to leave from system. Assume s to be the arrival rate of handover calls of type-2 at node N: Other assumptions of Model B is same as that in Model A. 3. Analysis 3.1. Exact solution of Model A Let jj ðtÞ represent the number of occupied channels of node j at time t; then the state of the tandem network at time t can be described as ðj1 ðtÞ; j2 ðtÞ; …; jN ðtÞÞ: Evidently, {ðj1 ðtÞ; j2 ðtÞ; …; jN ðtÞ; t $ 0} is a continuous Q time Markov process with finite state space, which contains Nj¼1 ðKj þ 1Þ possible states. Denote Pðn1 ; n2 ; …; nN Þ as the stationary probability distribution of this process, i.e. Pðn1 ;n2 ;…;nN Þ ¼ Pr{j1 ðtÞ ¼ n1 ; j2 ðtÞ ¼ n2 ;…; jN ðtÞ ¼ nN }lt!1 : Although one could write out the infinitesimal generator of this Markov process (see Example 1) and its balance
Fig. 3. A 4-node tandem network with a single-directional traffic flow.
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
211
Fig. 4. A 4-node tandem network with two-directional traffic flows.
equations, it is not easy to solve the system of equations because the number of equations is usually very large for large N: For example, if Kj ¼ K;;j ; then the complexity of this approach is OðK N Þ: So we shall develop an effective approach to get the approximate solution of system in Section 3.2.
Example 1. A 4-node tandem network with a unidirectional traffic flow (Fig. 3). Assume that N ¼ 4; K1 ¼ K2 ¼ K3 ¼ K4 ¼ 3 and m1 ¼ m2 ¼ m3 ¼ m4 ¼ 2: The four-dimensional Markov process {ðj1 ðtÞ; j2 ðtÞ; j3 ðtÞ; j4 ðtÞ; t $ 0} has 256 possible states. After all these states ðn1 ; n2 ; n3 ; n4 Þ are labeled in the lexicographic order. The infinitesimal generator Q1 of this process can be written as 3 2 0 0 2ðlþa1 ÞI64 ðlþa1 ÞI64 7 6 7 6 mC 2ðlþa1 þm1 ÞI64 ðlþa1 ÞI64 0 7 6 1 1 7 Q1 ¼ 6 7 6 7 6 0 2 m C 2ð l þ2 m ÞI l I 1 1 1 64 64 5 4 0
3m1 C1
0
23m1 I64
þI4 ^Q2 ; where 2
2a2 I16
a2 I16
0
3
0
7 6 7 6 m C 2ða þm ÞI a I 0 7 6 2 2 2 2 16 2 16 7þI4 ^Q3 ; 6 Q2 ¼ 6 7 7 6 0 2 m C 22 m I 0 2 2 2 16 5 4 0 0 3m2 C2 23m2 I16 2
2a3 I4
a 3 I4
0
0
3
7 6 6 m C 2ða þm ÞI a I 0 7 7 6 3 3 3 3 4 3 4 7þI4 ^Q4 ; Q3 ¼ 6 7 6 7 6 0 2 m C 22 m I 0 3 3 3 4 5 4 0 0 3m3 C3 23m3 I4 2
2 a4
6 6 m 6 4 Q4 ¼ 6 6 6 0 4 0
a4
0
0
2ða4 þ m4 Þ
a4
0
2m 4
22m4
0
0
3m 4
23m4
3 7 7 7 7; 7 7 5
2
p1
6 60 6 C1 ¼ 6 6 60 4 0 2
p2
6 60 6 C2 ¼ 6 6 60 4 0
0
3
0
7 07 7 7^I16 ; 7 q1 7 5 1
q2
0
0
3
p2
q2
0
p2
0
0
q1
0
p1
q1
0
p1
0
2
p3
7 6 60 07 7 6 7^I4 ; C3 ¼ 6 7 6 60 q2 7 5 4 1 0
q3
0
p3
q3
0
p3
0
0
0
3
7 07 7 7; 7 q3 7 5 1
where Ik is an identity matrix of order k:
3.2. Approximate solution of Model A At first let us decompose the system into N 2 1 pairs, in which Pair j ðj ¼ 1; …; N 2 1Þ consists of node j and node j þ 1: By the assumption of the model, we know there is no possibility of blocking occurring at any node j; j ¼ 1; …; N 2 1: It implies that ðj1 ðtÞ; j2 ðtÞ; …; jj ðtÞÞ is independent of ðjjþ1 ðtÞ; jjþ2 ðtÞ; …; jN ðtÞÞ: Thus when we calculate the stationary probability distribution of Pair j; we need not consider the state of node j þ 2; …; node N: But we have to incorporate the state of node j; j ¼ 1; …; j 2 1 into consideration, because ðjj ðtÞ; jjþ1 ðtÞÞ is dependent on ðj1 ðtÞ; j2 ðtÞ; …; jj21 ðtÞÞ: Our approximation method is based on the following two assumptions. Assumption 1. While we study the stationary distribution of Pair j; the process ðjj ðtÞ; jjþ1 ðtÞÞ is approximately seen as being dependent on jj21 ðtÞ only, and randomly independent of ðj1 ðtÞ; j2 ðtÞ; …; jj22 ðtÞÞ: Assumption 2. While we study the stationary distribution of Pair j; the arrival process of handover calls at node j; which come from node j 2 1; will be approximately described as a state-dependent Poisson process, i.e. whose arrival rate depends only on the state of node j:
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To justify the above assumptions, we give some intuitive explanation as follows. Although, as we said earlier, the stochastic behaviors of all up-stream nodes 1; …; j 2 1 can affect the state evolution of Pair j; the nodes 1; …; j 2 2 cannot affect Pair j without affecting node j 2 1 first, i.e. the affections from all up-stream nodes must be embodied at node j 2 1 before they function on Pair j: Thus the strongest dependency always lies between two neighboring nodes. Our Assumption 1 is made to capture this strongest part of stochastic correlations. Assumption 2 is based on following three facts. First, the handover calls’ arrival stream to node j is from the output stream of node j 2 1; the rate of which is completely determined by the state of node j 2 1: Second, the stochastic correlation between nodes j 2 1 and j is very strong because of their neighborhood, and the conditional probability distribution of state of node j 2 1 is obtainable when the state of node j is given. Third, the state-dependent Poisson process performs a very good behavior for fitting a general arrival process with the state-dependency. Besides, it should be mentioned that the approximate assumptions used in this paper have widely appeared in literature on analysis of queueing networks (see Ref. [3] and the reference therein for their justifications). Basic idea of our procedure is to calculate the stationary distribution of Pair j by aid of the knowledge of Pair j 2 1: The problem is how to find out a proper state-dependent Poisson process as an approximation of handover calls’ arrival process at node j: Let Pj ðnj ; njþ1 Þ denote the stationary distribution of the process ðjj ðtÞ; jjþ1 ðtÞÞ: Denote by Pj21 ðnj21 lnj Þ the conditional probability that there are nj21 occupied channels in node j 2 1 given that there are nj occupied channels in node j: Supposing that we have obtained the stationary distribution Pj21 ðnj21 ; nj Þ of Pair j 2 1; then we can calculate Pj21 ðnj21 lnj Þ: It is obvious that the output rate from node j 2 1 is nj21 mj21 given that there are nj21 occupied channels in node j 2 1: Hence, while we study the stationary distribution of the process ðjj ðtÞ; jjþ1 ðtÞÞ on Pair j; the average input rate of handover calls to node j can be approximately seen as X
Kj21
lj ðnj Þ¼
nj21 qj21 mj21 Pj21 ðnj21 lnj Þ; j¼2;3;…;N 21;
nj21 ¼1
which depends on state nj of node j only. Now let us focus our attention on Pair j where handover calls input to node j from node j 2 1 form a state-dependent Poisson process. This system can be described as a two-dimensional Markovian process with finite state space. After all states ðnj ; njþ1 Þ are labeled in the lexico~ j of the process graphic order, the infinitesimal generator Q
ðjj ðtÞ; jjþ1 ðtÞÞ can be written as 2
3
A~ ð0Þ B~ ð0Þ j j
7 6 7 6 ~ ~ ð1Þ ð1Þ 7 6 Cj Aj B~ j 7 6 7 6 7 6 ð2Þ ~ j A~ j 7 6 2 C 7 6 7 þIK þ1 ^R~ jþ1 ; ~ j ¼6 Q 7 6 j .. .. .. 7 6 7 6 . . . 7 6 7 6 6 ~ jðKj 21Þ B~ jðKj 21Þ 7 A 7 6 5 4 ðKj Þ Kj C~ j A~ j where ( aj ðiÞ ¼
A~ ðiÞ j ¼
aj ; if i , mj ;
j ¼ 1;2;…;N;
0; otherwise;
8 < 2ðlj ðiÞþaj ðiÞþimj ÞIKjþ1 þ1 ; if 0 # i # Kj 21; : 2Kj mj IK
jþ1 þ1
;
if i ¼ Kj ;
B~ ðiÞ j ¼ ðlj ðiÞþaj ðiÞÞIKjþ1 þ1 ; i ¼ 0;1;…;Kj 21; 2
pj mj qj mj ···
6 6 0 pm 6 j j 6 6 6 C~ j ¼ 6 6 6 6 0 6 0 4 0 0 2
··· ..
.
··· ···
0
0
3
7 0 7 7 7 7 7 7; 7 7 7 pj mj qj mj 7 5 0 mj 0
2ajþ1 ð0Þ ajþ1 ð0Þ 0 6 6 m 2ðmjþ1 þajþ1 ð1ÞÞ ajþ1 ð1Þ 6 jþ1 6 6 6 0 2mjþ1 2ð2mjþ1 þajþ1 ð2ÞÞ R~ jþ1 ¼ 6 6 6 6 6 4 0 0 0
···
0
···
0
···
0
..
.
··· 2Kjþ1 mjþ1
3 7 7 7 7 7 7 7; 7 7 7 7 5
C~ j and R~ jþ1 are matrices of order Kjþ1 þ1; and Ik is an identity matrix of order k: After labelling all of Pj ðnj ; njþ1 Þ in the lexicographic order, we obtain a column vector of dimension ðKj þ 1ÞðKjþ1 þ 1Þ; which is denoted by p~ j : It can be solved by the equations 8 ~ j ¼ 0; < p~ Tj Q : p~ T e ¼ 1; j
where e is a column vector of dimension ðKj þ 1ÞðKjþ1 þ 1Þ and all of its elements are 1. The following simple recursive algorithm can be used to obtain all Pj ðnj ; njþ1 Þ; j ¼ 1; 2; …; N 2 1:
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
Step 1 Note that l1 ðn1 Þ ¼ l þ a1 for 0 # n1 , m1 ; otherwise l1 ðn1 Þ ¼ l for m1 # n1 # K1 : Thus for Pair 1 we can write out its state equations and obtain the exact solution of P1 ðn1 ; n2 Þ and P1 ðn1 ln2 Þ: P Kj21 Step 2 For jð2 # j # N 2 1Þ; letting lj ðnj Þ ¼ nj21 ¼1 nj21 qj21 mj21 Pj21 ðnj21 lnj Þ; we use the balance equations of Pair j to calculate Pj ðnj ; njþ1 Þ: If j , N 2 1; then we further calculate Pj ðnj lnjþ1 Þ and return to Step 2 after setting j ¼ j þ 1; otherwise stop. Again, if we let Kj ¼ K; ;j ; then the complexity of our approximation algorithm is OðNK 2 Þ: Compared with OðK N Þ; this approximation is a considerable saving, provided that the associated error is very small. 3.3. Performance measures of Model A The loss probability of handover calls at node j ðj ¼ 2; …; NÞ is the probability of a handover call finding no channel available in node j; while it leaves from node j 2 1 and intends making connection to node j: Thus P{A handover call being lost at node j} 8 K2 X > > > P1 ðK1 ; n2 Þ; if j ¼ 1; > > > > n2 ¼0 > > > > < KX j21 ¼ nj21 mj21 qj21 Pj21 ðnj21 ; Kj Þ > > nj21 ¼0 > > ; if j ¼ 2;…; N: > > Kj K j21 > X X > > > nj21 mj21 qj21 Pj21 ðnj21 ;nj Þ > : nj ¼0 nj21 ¼0
By Poisson arrival see time average (PASTA), we know that the loss probability of fresh calls is the probability that the number of occupied channels in a node is equal to or greater than the specific control threshold value, mj : So P{A fresh call being lost at node j} 8 K2 K 1 X X > > > P1 ðn1 ; n2 Þ; if j ¼ 1; > > < n2 ¼0 n1 ¼m1 ¼ Kj Kj21 > > X X > > Pj21 ðnj21 ;nj Þ; if j ¼ 2; …;N: > : nj ¼mj nj21 ¼0
The expected number of occupied channels at node j is 8 K2 K1 X X > > > n1 P1 ðn1 ;n2 Þ; if j ¼ 1; > > < n2 ¼0 n1 ¼0 Ejj ¼ Kj Kj21 > > X X > > nj Pj21 ðnj21 ; nj Þ if j ¼ 2;…; N: > : nj ¼0 nj21 ¼0
213
3.4. Exact solution of Model B Let jj ðtÞ ¼ ðj1;j ðtÞ; j2;j ðtÞÞ; where j1;j ðtÞ and j2;j ðtÞ represent the numbers of channels occupied by type-1 and type2 in node j at time t; respectively. So j1;j ðtÞ $ 0; j2;j ðtÞ $ 0 and j1;j ðtÞ þ j2;j ðtÞ # Kj : The state of this tandem network at time t can be described as ðj1 ðtÞ; j2 ðtÞ; …; jN ðtÞÞ: Obviously, {ðj1 ðtÞ; j2 ðtÞ; …; jN ðtÞ; t $ 0} is a finite state Markov process, whose state space contains Q 22N Nj¼1 ðKj þ 1ÞðKj þ 2Þ possible states. From the theoretical point of view, we can write out the infinitesimal generator of this Markov process (refer to Example 2) and its balance equations. However, the number of balance equations increases rapidly as N increases. For instance, if Kj ¼ K; ;j ; the complexity of solving the equations is Oðð 12 K 2 ÞN Þ: Hence developing an effective approximation becomes our main task in Section 3.5. Example 2. A 3-node tandem network with two-directional flows (Fig. 4). Assume that N ¼ 3; K1 ¼ K3 ¼ 2; K2 ¼ 3; m1 ¼ m3 ¼ 1 and m2 ¼ 2: The Markov process ðj1 ðtÞ; j2 ðtÞ; j3 ðtÞÞ; t $ 0} has 360 possible states. Let us label all states of j1 ðtÞ in the order ð0; 0Þ; ð0; 1Þ; ð0; 2Þ; ð1; 0Þ; ð1; 1Þ; ð2; 0Þ: The similar work can be done for all states of j2 ðtÞ and j3 ðtÞ; respectively. After that, all possible states of ðj1 ðtÞ; j2 ðtÞ; j3 ðtÞÞ are labeled in the lexicographic order. Then the infinitesimal generator Q of this process can be written as
2
3
2
Að0Þ Bð0Þ 2 2
0
0
3
7 6 0 7 6 ð1Þ ð1Þ ð1Þ 7 6 C2 A2 B2 6 7 0 7 6 6 ð1Þ ð1Þ ð1Þ 7 Q ¼ 6 C1 A1 B1 7 þ I6 ^6 7 6 0 C ð2Þ Að2Þ Bð2Þ 7 4 5 7 6 2 2 2 5 4 0 C1ð2Þ Að2Þ 1 ð3Þ ð3Þ 0 0 C2 A2 3 2 ð0Þ D2 0 0 0 7 6 7 6 7 6 0 Dð1Þ 0 0 2 7 6 þ H1 ^6 7 7 6 0 ð2Þ 0 D 0 7 6 2 5 4 ð3Þ 0 0 0 D2 2 ð0Þ ð0Þ 3 2 ð0Þ 3 0 0 0 A3 B 3 D3 6 7 6 7 6 6 7 ð1Þ 7 þ I60 ^6 C3ð1Þ Að1Þ 7 þ H2 ^6 0 Dð1Þ 7; B 0 3 3 3 4 5 4 5 ð2Þ 0 0 D 0 C3ð2Þ Að2Þ 3 3 Að0Þ 1
Bð0Þ 1
where 2 6 6 Að0Þ 1 ¼4
2ðl þ a1 þ b1 Þ
b1
0
m2;1
2ðl þ m2;1 Þ
0
0
2m2;1
22m2;1
3 7 7^I60 ; 5
214
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
"
Að1Þ 1 ¼
0
m2;1
2ðm1;1 þ m2;1 Þ
Að2Þ 1 ¼ 22m1;1 I60 ; 2 3 l þ a1 0 6 7 6 0 Bð0Þ l7 1 ¼4 5^I60 ;
C1ð1Þ ¼
"
0
m1;1
6 6 6 6 6 6 6 6 6 6 6 6 6 J2 ¼ 6 6 6 6 6 6 6 6 6 6 6 6 6 4
0
0
p1;1
0
0
0
0
p1;1 0
0
0
0
1
0
0
0
0 p1;1
0
0
0
0
0
0
0
0
0
0 0
2
^I60 ;
0
0
0
0
0
q1;1
0
0
0
0
0
q1;1
0
0
0
0
0
0
0
0
0
q1;1
0
0
p1;1
0
0
q1;1
0
0
0
1
0
0
0
0
0
0
0
p1;1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
3
7 0 7 7 7 7 0 7 7 7 0 7 7 7 0 7 7 7^I6 ; 7 0 7 7 7 0 7 7 7 7 q1;1 7 7 7 0 7 5 1 3
b2
0
0
0
2ða2 þ b2 þ m2;2 Þ
b2
0
0
0
22m2;2
0
0
0
0
23m2;2
6 4
7 7 7 7^I6 ; 7 7 5
b2
0
0
2ðm1;2 þ m2;2 Þ
0
0
0
2ðm1;2 þ2m2;2 Þ
22m1;2
0
0
2ð2m1;2 þ m2;2 Þ
a2 0 0
6 6 6 0 a2 ð0Þ 6 B2 ¼ 6 6 60 0 4
7 7^I6 ; 5
Að3Þ 2 ¼ 23m1;2 I6 ;
^I6 ;
3
2 7 a2 7 6 07 7 6 0 7^I6 ; Bð1Þ 2 ¼6 7 4 07 5 0
0
6 C2ð1Þ ¼ 6 4
m1;2 0
0 0
3
"
0
6 6 6 6 6 6 J3 ¼ 6 6 6 6 6 6 4
0
0
0
p2;2
q2;2
0
0
0
0
1
0
0
0
0
0
p2;2
q2;2
0
0
0
0
1
0
0
0
0
0
p1;2
0
0
q1;2
0
0
p1;2
0
0
q1;2
0
0
1
0
0
0
0
0
p1;2
0
0
0
0
0
1
0
0
0
0
0
6 6 Að0Þ 3 ¼4 2 4 Að1Þ 3 ¼
#
2m1;2 0 0 7 ð2Þ 0 m1;2 0 0 7 ^J3 ; 5^J3 ; C2 ¼ 0 2m1;2 0 0 0 m1;2 0
7 07 7 7^I6 ; 7 07 5 0
0
3
7 07 7 7 07 7 7; 7 07 7 7 07 5 0
1 3
7 0 7 7 7 0 7 7 7; 7 q1;2 7 7 7 0 7 5 1
b3 þ s
0
0
2ðm2;3 þ sÞ
s
0
0
22m2;3
s
0 3 0 7 7 0 7; 5
2ðm1;3 þ m2;3 Þ
0
3 7 7; 5
3
2ðm1;3 þ sÞ
2 3 0 4 5; Bð1Þ ¼ 3 0
Dð0Þ 3
3
2ða3 þ b3 þ sÞ
a3 6 6 Bð0Þ 0 3 ¼6 4
2
0
" # 0 0 7 ð2Þ 7 0 5^I6 ; D2 ¼ ^I6 ; Dð3Þ 2 ¼0·I6 ; m2;2 0 0
0
2
0
3
q2;2
6 6 6 6 6 6 H1 ¼ 6 6 6 6 6 6 4 2
0
p2;2
0
0 0 0 2
2
3
2 3 7 0 7 ð2Þ 4 5 0 7^I6 ; B2 ¼ ^I6 ; 5 0 0
0
0 2m2;2
2
#
0
0
6 6m 0 6 2;2 0 6 Dð0Þ ¼ 6 2 6 0 2m2;2 0 4 0 0 3m2;2
0 ^J3 ;
3
2ða2 þ b2 þ m1;2 Þ
2
0
6 6 Dð1Þ 2 ¼ 4 m2;2
2ða2 þ b2 Þ
6 6 Að0Þ 2 ¼6
"
0
C1ð2Þ ¼ 2m1;1 0 ^J2 ;
^J2 ; 0 q1;1
6 6 Að1Þ 2 ¼4
¼
" # l
0
2
C2ð3Þ ¼ 3m1;2
#
0
6 6
^I60 ;
Bð1Þ 1
p1;1
2
Að2Þ 2 ¼
0
m1;1 0
0
2
0
2
#
2ðl þ m1;1 Þ
5;
Að2Þ 3 ¼ 22m1;3 ;
0 2 C3ð1Þ ¼ 4
0
0
m1;3 0
0
0
m1;3 0
3
6 7 0 07 ¼6 4 m2;3 5; 0 2m2;3 0
Dð1Þ 3
3 5;
" ¼
0
C3ð2Þ ¼ 2m1;3 0 ;
0
m2;3 0
# ;
Dð2Þ 3 ¼ 0;
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
2 6 6 6 6 6 6 6 6 6 6 6 6 6 H 2 ¼ I6 ^6 6 6 6 6 6 6 6 6 6 6 6 6 4
p2;3 q2;3 0
0
p2;3 q2;3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
p2;3 q2;3
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
p2;3 q2;3
p2;3 q2;3
p2;3 q2;3
0
3
7 07 7 7 7 07 7 7 07 7 7 07 7 7: 7 07 7 7 07 7 7 7 07 7 7 07 5 1
3.5. Approximate solution of Model B Similar to Model A, we decompose the system into N 2 1 pairs, in which Pair j ðj ¼ 1; …; N 2 1Þ consists of node j and node j þ 1: Compared with the uni-directional traffic model, there are stronger stochastic dependency between two neighbor pairs in our Model B. Namely, the process ðjj ðtÞ; jjþ1 ðtÞÞ depends randomly on both ðj1 ðtÞ; j2 ðtÞ; …; jj21 ðtÞÞ and ðjjþ2 ðtÞ; jjþ3 ðtÞ; …; jN ðtÞÞ: The following two assumptions are the starting points of our approximation idea. Assumption 3. While we study the stationary distribution of Pair j; the process ðjj ðtÞ; jjþ1 ðtÞÞ is approximately seen as being only dependent on jj21 ðtÞÞ and jjþ2 ðtÞÞ; and randomly independent of ðj1 ðtÞ; j2 ðtÞ; …; jj22 ðtÞÞ and ðjjþ3 ðtÞ; jjþ4 ðtÞ; …; jN ðtÞÞ: Assumption 4. While we study the stationary distribution of Pair j; the arrival process of type-1 handover calls to node j from node j 2 1 and the arrival process of type-2 handover calls to node j þ 1 from node j þ 2 will be approximately described as two state-dependent Poisson processes, whose arrival rates depend on the number of occupied channels in node j only. An explanation for the justification of these two assumptions is similar to that in Model A. Our approach is to calculate the stationary distribution of Pair j by using the knowledge of Pair j 2 1 and Pair j þ 1: The key problem is to obtain two state-dependent Poisson processes for node j to fit arrival processes of type-1 and type-2 handover calls from node j 2 1 and node j þ 2; respectively, so as to capture the stochastic correlation between Pair j and all upstream and downstream nodes. Let Pj ðn1;j ; n2;j ; n1;jþ1 ; n2;jþ1 Þ denote the stationary distribution of the process ðjj ðtÞ; jjþ1 ðtÞÞ; j ¼ 1; 2; …; N 2 1: Define nj ¼ n1;j þ n2;j ; j ¼ 1; 2; …; N; which denote the total number of occupied channels in node j: Denote Pj21 ðn1;j21 lnj Þ as the conditional probability
215
that there are n1;j21 channels occupied by type-1 calls in node j 2 1 given that there are totally nj occupied channels in node j; and denote Pjþ1 ðn2;jþ2 lnjþ1 Þ by the conditional probability that there are n2;jþ2 channels occupied by type-2 calls in node j þ 2 given that there are totally njþ1 occupied channels in node j þ 1: Supposing that Pj21 ðn1;j21 ; n2;j21 ; n1;j ; n2;j Þ and Pjþ1 ðn1;jþ1 ; n2;jþ1 ; n1;jþ2 ; n2;jþ2 Þ have been obtained, then we can calculate Pj21 ðn1;j21 lnj Þ and Pjþ1 ðn2;jþ2 lnjþ1 Þ easily. It is obvious that the output rate of type-1 calls from node j 2 1 is n1;j21 m1;j21 given that there are n1;j21 channels occupied by type-1 calls in node j 2 1; and the output rate of type-2 calls from node j þ 2 is n2;jþ2 m2;jþ2 given that there are n2;jþ2 channels occupied by type-2 calls in node j þ 2: Hence, while we study the stationary distribution of the process ðjj ðtÞ; jjþ1 ðtÞÞ; the conditional arrival rate of type-1 handover calls to node j can be approximately seen as
lj ðnj Þ ¼
X
q1;j21 n1;j21 m1;j21 Pj21 ðn1;j21 lnj Þ;
0#n1;j21 #Kj21
j ¼ 2; …; N 2 1; and the conditional arrival rate of type-2 handover calls to node j þ 1 can be approximately seen as X sjþ1 ðnjþ1 Þ ¼ q2;jþ2 n2;jþ2 m2;jþ2 Pjþ1 ðn2;jþ2 lnjþ1 Þ; 0#n2;jþ2 #Kjþ2
j ¼ N 2 2; …1: In other words, lj ðnj Þ (or sjþ1 ðnjþ1 Þ) is the approximate rate of type-1 (or type-2) handover calls going to node j (or j þ 1) from node j 2 1 (or j þ 2) given that there are nj (or njþ1 ) channels occupied in node j (or j þ 1) totally. Now let us focus our attention on Pair j: Define ( bj ; if i , mj ; bj ðiÞ ¼ 0; otherwise; ( lj ðiÞ þ aj ðiÞ; if 0 # i # Kj 2 1; cj ðiÞ ¼ 0; if i ¼ Kj ; ( sj ðiÞ þ bj ðiÞ; if 0 # i # Kj 2 1; dj ðiÞ ¼ 0; if i ¼ Kj ; ( lj ðiÞ þ aj ðiÞ þ bj ðiÞ; if 0 # i # Kj 2 1; ej ðiÞ ¼ 0; if i ¼ Kj ; ( sj ðiÞ þ aj ðiÞ þ bj ðiÞ; if 0 # i # Kj 2 1; fj ðiÞ ¼ 0; if i ¼ Kj : aj ðiÞ has same definition as that in Section 3.2. All possible states of node j are denoted by ðn1;j ; n2;j Þ; which means that there are n1;j channels occupied by type-1 calls and n2;j channels occupied by type-2 calls. Let us label
216
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
these states in following order: ð0; 0Þ; ð0;1Þ;…; ð0;Kj Þ;ð1; 0Þ; ð1;1Þ; …;ð1;Kj 2 1Þ;…; ðKj ; 0Þ: After all the states of Pair j (node j and node j þ 1) are labeled in the lexicographic order, the infinitesimal ~ j of the process ðjj ðtÞ; jjþ1 ðtÞÞ can be written as generator Q 2
3
A~ ð0Þ B~ ð0Þ j j
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
6 6 ð1Þ ð1Þ ð1Þ 6 C~ 6 j A~ j B~ j 6 6 6 .. 6 ~ Qj ¼6 C~ ð2Þ A~ ð2Þ . j j 6 6 6 6 .. .. 6 . . 6 4
B^ ðmÞ is ðKj 2 m þ 1Þ £ ðKj 2 mÞ matrix j 3 2 ð0Þ ð0Þ ^ E^ 3 2 D jþ1 jþ1 m1;j 0 7 6 7 6 7 6 . 7 6 7 6 7 6 m1;j 0 ^ ð1Þ . . D 7 6 jþ1 7 6 7 6 ðmÞ 7; 6 ^ ^ ;J ¼ Cj ¼m·6 7 .. .. 7 7 jþ1 6 6 . 7 6 ðK 21Þ 7 6 jþ1 . . . ^ 7 6 . E 5 4 jþ1 7 6 5 4 m1;j 0 ðKjþ1 Þ ^ D jþ1
ðK Þ A~ j j þ I 1 ðK þ1ÞðK þ2Þ ^R^ 0jþ1 þ H^ j ^R^ 00jþ1 ; 2
j
is ðKj 2 m þ 1Þ £ ðKj 2 m þ 1Þ matrix A^ ðmÞ j 3 2 cj ðmÞ 7 6 7 6 0 cj ðm þ 1Þ 7 6 7 6 7 6 . 7 6 ðmÞ .. B^ j ¼ 6 7; 0 7 6 7 6 .. 7 6 7 6 . c ðK 2 1Þ j j 5 4 0
j
where 2
3
^ ð0Þ V^ ð0Þ U jþ1 jþ1
6 6 ^ ð1Þ ^ ð1Þ ^ ð1Þ 6 Wjþ1 Ujþ1 Vjþ1 6 6 6 . ^R0jþ1 ¼ 6 ^ ð2Þ U ^ ð2Þ . . 6 W jþ1 jþ1 6 6 6 .. .. 6 6 . . 4 2 6 6 6 6 00 R^ jþ1 ¼ 6 6 6 6 4
ðK Þ U^ jþ1jþ1
2
7 7 7 7 7 7 7 7; 7 7 7 7 7 5
6 6 6 6 ðmÞ ^ ¼6 D jþ1 6 6 6 4
Y^ ð1Þ jþ1 ..
. ðK Þ Y^ jþ1jþ1
7 7 7 7 7; 7 7 7 5
^ ðmÞ A~ ðmÞ j ¼ Aj ^I 12 ðKjþ1 þ1ÞðKjþ1 þ2Þ ;
0 # m # Kj ;
^ ðmÞ B~ ðmÞ j ¼ Bj ^I 12 ðKjþ1 þ1ÞðKjþ1 þ2Þ ;
0 # m # Kj 2 1;
^ ðmÞ ^ C~ ðmÞ j ¼ Cj ^Jjþ1 ; 2
2ej ðmÞ
6 6 m 6 2;j 6 6 6 6 A^ ðmÞ j ¼6 6 6 6 6 4
3
bj ðmÞ 2ej ðm þ 1Þ bj ðm þ 1Þ
..
.
2ej ðm þ 2Þ
3
q1;j
7 7 6 7 7 6 . 7 7 6 . 7 ðmÞ 6 0 . 7 . 7; E^ ¼6 7; 7 jþ1 6 . 7 7 7 6 . 7 6 p1;j 7 . q 1;j 5 5 4 0 1
2m2;jþ1 ·diagð0;1;…;Kjþ1 2mÞ2mm1;jþ1 ·IKjþ1 2mþ1 ;
1 # m # Kj ;
2m2;j
..
2
^ ðmÞ ð0#m#Kjþ1 Þ are C^ ðmÞ is ðKj 2mÞ£ðKj 2mþ1Þ matrix, D j jþ1 ðKjþ1 2mþ1Þ£ðKjþ1 2mþ1Þ matrix, and E^ ðmÞ jþ1 ð0#m# Kjþ1 21Þ are ðKjþ1 2mþ1Þ£ðKjþ1 2mÞ matrix 3 2 2fjþ1 ðmÞ djþ1 ðmÞ 7 6 7 6 7 6 2fjþ1 ðmþ1Þ djþ1 ðmþ1Þ 7 6 7 6 7 6 .. 7 6 ðmÞ 7 2f ðmþ2Þ . U^ jþ1 ¼ 6 jþ1 7 6 7 6 7 6 7 6 . .. 7 6 7 6 5 4 2fjþ1 ðKjþ1 Þ
3
Y^ ð0Þ jþ1
3
p1;j
..
.
..
. 2ej ðKj Þ
2 m2;j ·diagð0;1; …;Kj 2 mÞ 2 mm1;j ·IKj 2mþ1 ;
7 7 7 7 7 7 7 7 7 7 7 7 5
2
3
0
7 7 7 7 7 7 7 7; 7 7 7 7 7 5
6 6 6 m2;jþ1 0 6 6 6 6 2m2;jþ1 0 Y^ ðmÞ jþ1 ¼ 6 6 6 6 .. .. 6 . . 6 4 0
^ ðmÞ U^ ðmÞ jþ1 and Yjþ1 are ðKjþ1 2 m þ 1Þ £ ðKjþ1 2 m þ 1Þ matrix
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
ð0 # m # Kjþ1 Þ 2 ajþ1 ðmÞ 6 6 0 ajþ1 ðm þ 1Þ 6 6 6 6 V^ ðmÞ 0 jþ1 ¼ 6 6 6 6 6 4
The following recursive algorithm can be used to obtain Pj ðn1;j ; n2;j ; n1;jþ1 ; n2;jþ1 Þ for all j ¼ 1; 2; …; N 2 1:
3 7 7 7 7 7 .. 7 7; . 7 7 .. 7 . ajþ1 ðKjþ1 2 1Þ 7 5 0
V^ ðmÞ jþ1 are ðKjþ1 2 m þ 1Þ £ ðKjþ1 2 mÞ matrix ð0 # m # Kjþ1 2 1Þ 3 2 m1;jþ1 0 7 6 7 6 m1;jþ1 0 7 6 7 6 ðmÞ ^ W 7; jþ1 ¼ m·6 . . 7 6 . . 7 6 . . 5 4 m1;jþ1 0 ^ ðmÞ are ðKjþ1 2 mÞ £ ðKjþ1 2 m þ 1Þ matrix ð1 # m # Kjþ1 Þ W jþ1 2 6 6 6 6 ^ Hj ¼ 6 6 6 6 4
3
F^ ð0Þ j
2 6 6 6 6 ^FðmÞ j ¼6 6 6 4
F^ ð1Þ j ..
. ðK Þ F^ j j
7 7 7 7 7 7 7 7 5 3
p2;jþ1 q2;jþ1 ..
.
..
.
p2;jþ1
Step 1 Select the initial approximations for sj ðnj Þ; 0 # nj # Kj and j ¼ 2; …; N 2 1: Step 2 We take Step 2.1 –Step 2.4 as the kth iteration ðk ¼ 1; 2; …Þ: Step 2.1. Noting l1 ðn1 Þ ¼ l þ a1 for 0 # n1 , m1 ; otherwise l1 ðn1 Þ ¼ l for m1 # n1 # K1 ; thus for Pair 1 we can write out its balance equations and obtain its solution of P1 ðn1;1 ; n2;1 ; n1;2 ; n2;2 Þ and then calculate l2 ðn2 Þ; 0 # n2 # K2 : Step 2.2. For jð2 # j # N 2 2Þ; we use the balance equations of Pair j to calculate Pj ðn1;j ; n2;j ; n1;jþ1 ; n2;jþ1 Þ and calculate ljþ1 ðnjþ1 Þ; 0 # njþ1 # Kjþ1 : After setting j ¼ j þ 1; repeat Step 2.2 if j # N 2 2; otherwise take Step 2.3. Step 2.3. Noting sN ðnN Þ ; s for all 0 # nN # KN ; thus for Pair N 2 1 we can write out its state equations and obtain its solution of PN21 ðn1;N21 ; n2;N21 ; n1;N ; n2;N Þ and new values of sN21 ðnN21 Þ; 0 # nN21 # KN21 : Step 2.4. For jðN 2 1 $ j $ 3Þ; we use the balance equations of Pair j 2 1 to calculate Pj21 ðn1;j21 ; n2;j21 ; n1;j ; n2;j Þ and then calculate new values of sj21 ðnj21 Þ; 0 # nj21 # Kj21 : After setting j ¼ j 2 1; repeat Step 2.4 if j $ 3; otherwise go back to Step 2 (starting a new iteration) after setting k ¼ k þ 1: Stopping condition for the iterations. The iterations will be terminated if a convergence criterion has been satisfied (e.g. the maximum absolute value of probability distributions’ differences between two iterations for each pair less than a given value).
7 7 7 7 7; 7 q2;jþ1 7 5 1
3.6. Performance measures of Model B
F^ ðmÞ j ð0 # m # Kj Þ
are ðKj 2 m þ 1Þ £ ðKj 2 m þ 1Þ matrix. After arranging all of Pj ðn1;j ; n2;j ; n1;jþ1 ; n2;jþ1 Þ in the lexicographic order, we obtain a column vector of dimension 14 ðKj þ 1ÞðKj þ 2ÞðKjþ1 þ 1ÞðKjþ1 þ 2Þ; which is denoted by p ~ j: It can be solved for by the equations 8 ~ j ¼ 0;
For j ¼ 1; 2; …; N; we define that
Qj ¼ {ðn1;j ; n2;j Þln1;j þ n2;j # Kj ; n1;j $ 0; n2;j $ 0}; Dj ¼ {ðn1;j ; n2;j Þln1;j þ n2;j ¼ Kj ; n1;j $ 0; n2;j $ 0}; Fj ¼ {ðn1;j ; n2;j Þlmj # n1;j þ n2;j # Kj ; n1;j $ 0; n2;j $ 0}:
j
where e is an identity column vector of dimension 1ÞðKj þ 2ÞðKjþ1 þ 1ÞðKjþ1 þ 2Þ:
217
1 4
ðKj þ
There are two types of handover call losses corresponding to type-1 and type-2 handover calls, respectively.
8X X P ðn ; n ; n ; n Þ; > > < Q2 D1 1 1;1 2;1 1;2 2;2 P P P{A type-1 handover call is lost at node j} ¼ Dj Qj21 n1;j21 m1;j21 q1;j21 Pj21 ðn1;j21 ; n2;j21 ; n1;j ; n2;j Þ > > ; :P P Qj Qj21 n1;j21 m1;j21 q1;j21 Pj21 ðn1;j21 ; n2;j21 ; n1;j ; n2;j Þ
if j ¼ 1; if j ¼ 2; …; N:
218
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
8 P P Qjþ1 Dj n2;jþ1 m2;jþ1 q2;jþ1 Pj ðn1;j ; n2;j ; n1;jþ1 ; n2;jþ1 Þ > > ;
X X > : PN21 ðn1;N21 ; n2;N21 ; n1;N ; n2;N Þ; DN Q
if j ¼ 1; …; N 2 1; if j ¼ N;
N21
Both type-1 fresh calls and type-2 fresh calls have the same loss probability, which is the probability of the number of occupied channels in a node being equal to or greater than the specific control threshold value (by PASTA). P{A fresh call is lost at node j} 8X X > P1 ðn1;1 ; n2;1 ; n1;2 ; n2;2 Þ; > > < Q2 F1 ¼ X X > > Pj21 ðn1;j21 ; n2;j21 ; n1;j ; n2;j Þ; > :
if j ¼ 1; if j ¼ 2; …; N:
Fj Qj21
The expected number of occupied channels in node j is Eðj1;j þ j2;j Þ 8X X ðn1;1 þn2;1 ÞP1 ðn1;1 ;n2;1 ;n1;2 ;n2;2 Þ; if j ¼ 1; > > > Q2 Q1 < ¼ XX > > ðn1;j þn2;j ÞPj21 ðn1;j21 ;n2;j21 ;n1;j ;n2;j Þ; if j ¼ 2;…;N: > : Qj Qj21
4. Comparison of numerical results There are two ways of assessing our approximations. One is to use simulation results to compare with the approximations. The other is to work with some small examples and obtain their exact solutions, which can be compared with the approximations. We choose the latter because the comparison is easier to make. We do not use simulation results for comparison because simulation in
itself is a sampling procedure, it might not be good for assessing the quality of our approximation. To illustrate the performance of our method, we have tested our approximation by two examples in a wide range of parameter settings. In order to get the exact results as the contrasts of our approximate results, we use Examples 1 and 2 in Sections 3.1 and 3.4 for testing. Their exact solutions are obtained by solving their global balance equations numerically. The performance indexes for comparison are the loss probabilities of handover calls and fresh calls, and the expected number of occupied channels. The results of the comparisons are reported in Tables 1– 8, in which Tables 1 –4 and Tables 5 –8 are on Model A and Model B, respectively. For Model B, our approach requires the iterative calculations. The condition for stopping the iterations is that a maximum difference of probability distributions for all pairs between two consecutive iterations is less than 1026. In the headers of these tables, the abbreviations Exa., App. and Err.% mean the exact solution, the approximate solution and its relative error, respectively, and † Loss Handover Call V the loss probability of handover calls, † Loss Fresh Call V the loss probability of fresh calls, † Exp. Ocp. Ch. V the expected number of occupied channels, † Loss Handover-i V the loss probability of handover calls of type-i; ði ¼ 1; 2Þ; † Loss Handover Call V the loss probability of handover calls,
Table 1 Comparison of exact and approximate results for different l in Example 1 (Model A) m ¼ ð1; 1; 1; 1Þ; a ¼ ð0:8; 1; 0:5; 1Þ; p ¼ ð0:1; 0:3; 0:2Þ
l
0.5
3
10
n
1 2 3 4 1 2 3 4 1 2 3 4
Loss Handover Call
Loss Fresh Call
Exp. Ocp. Ch.
Exa.
App.
Err.%
Exa.
App.
Err.%
Exa.
App.
Err.%
0.0429 0.0842 0.0662 0.0893 0.3753 0.2115 0.1045 0.1080 0.7349 0.2977 0.1223 0.1149
0.0429 0.0842 0.0676 0.0915 0.3753 0.2115 0.1066 0.1104 0.7349 0.2977 0.1231 0.1170
0.0000 0.0000 2.2425 2.4945 0.0000 0.0000 2.0337 2.2042 0.0000 0.0000 0.6485 1.8830
0.3000 0.4628 0.3757 0.4737 0.7505 0.6521 0.4603 0.5097 0.9554 0.7177 0.4883 0.5210
0.3000 0.4628 0.3752 0.4727 0.7505 0.6521 0.4593 0.5083 0.9554 0.7177 0.4879 0.5199
0.0000 0.0000 0.1258 0.2103 0.0000 0.0000 0.2227 0.2741 0.0000 0.0000 0.0884 0.2035
1.0385 1.3932 1.2228 1.4172 2.0738 1.8196 1.4105 1.4968 2.6865 1.9804 1.4725 1.5217
1.0385 1.3932 1.2216 1.4151 2.7038 1.8196 1.4083 1.4939 2.6865 1.9804 1.4716 1.5196
0.0000 0.0000 0.0990 0.1456 0.0000 0.0000 0.1555 0.1905 0.0000 0.0000 0.0600 0.1388
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
219
Table 2 Comparison of exact and approximate results for different m in Example 1 (Model A)
l ¼ 3; a ¼ ð0:5; 0:5; 0:5; 0:5Þ; p ¼ ð0:1; 0:1; 0:1Þ mT
n
Loss Handover Call
Loss Prob. Fresh Call
Exp. Ocp. Ch.
Exa.
App.
Err.%
Exa.
App.
Err.%
Exa.
App.
Err.%
1 1 1 1
1 2 3 4
0.3657 0.1847 0.1366 0.1178
0.3657 0.1847 0.1397 0.1214
0.0000 0.0000 2.2672 3.0803
0.7313 0.5903 0.5269 0.4958
0.7313 0.5903 0.5250 0.4928
0.0000 0.0000 0.3565 0.6223
2.0373 1.6997 1.5573 1.4886
2.0373 1.6997 1.5535 1.4820
0.0000 0.0000 0.2438 0.4397
1 10 1 10
1 2 3 4
0.3657 0.0004 0.2316 0.0003
0.3657 0.0004 0.2447 0.0003
0.0000 0.0000 5.6572 3.4849
0.7313 0.0205 0.6298 0.0170
0.7313 0.0205 0.6190 0.0168
0.0000 0.0000 1.7209 1.1418
2.0373 0.2323 1.7914 0.2103
2.0373 0.2323 1.7695 0.2084
0.0000 0.0000 1.2260 0.9356
10 1 10 1
1 2 3 4
0.0043 0.3764 0.0004 0.2353
0.0043 0.3764 0.0004 0.2483
0.0000 0.0000 0.4710 5.5330
0.0475 0.7452 0.0210 0.6344
0.0475 0.7452 0.0210 0.6235
0.0000 0.0000 0.0208 1.7201
0.3463 2.0710 0.2353 1.8020
0.3463 2.0710 0.2353 1.7799
0.0000 0.0000 0.0002 1.2272
Table 3 Comparison of exact and approximate results for different p in Example 1 (Model A)
l ¼ 4; ; m ¼ ð0:8; 0:8; 0:8; 0:8Þ; a ¼ ð1; 1; 1; 1Þ pT
n
Loss Handover Call
Loss Fresh Call
Exp. Ocp. Ch.
Exa.
App.
Err.%
Exa.
App.
Err.%
Exa.
App.
Err.%
0.05 0.05 0.05
1 2 3 4
0.5486 0.2802 0.2198 0.2031
0.5486 0.2802 0.2215 0.2054
0.0000 0.0000 0.7552 1.1437
0.8778 0.7266 0.6803 0.6651
0.8778 0.7266 0.6793 0.6633
0.0000 0.0000 0.1429 0.2577
2.4096 1.9894 1.8741 1.8375
2.4096 1.9894 1.8722 1.8341
0.0000 0.0000 0.1026 0.1874
0.5 0.5 0.5
1 2 3 4
0.5486 0.1388 0.0774 0.0647
0.5486 0.1388 0.0780 0.0653
0.0000 0.0000 0.6768 0.9533
0.8778 0.5570 0.4699 0.4465
0.8778 0.5570 0.4698 0.4463
0.0000 0.0000 0.0278 0.0376
2.4096 1.5913 1.3967 1.3450
2.4096 1.5913 1.3964 1.3447
0.0000 0.0000 0.0182 0.0253
0.95 0.95 0.95
1 2 3 4
0.5486 0.0098 0.0034 0.0032
0.5486 0.0098 0.0034 0.0032
0.0000 0.0000 0.0179 0.0100
0.8778 0.2919 0.2720 0.2714
0.8778 0.2919 0.2720 0.2714
0.0000 0.0000 0.0000 0.0000
2.4096 1.0045 0.9600 0.9586
2.4096 1.0045 0.9600 0.9586
0.0000 0.0000 0.0000 0.0000
Table 4 Comparison of exact and approximate results for different a in Example 1 (Model A)
l ¼ 5; m ¼ ð2; 2; 2; 2Þ; p ¼ ð0:1; 0:1; 0:1Þ aT
n
Loss Handover Call
Loss Fresh Call
Exp. Ocp. Ch.
Exa.
App.
Err.%
Exa.
App.
Err.%
Exa.
App.
Err.%
0.1 0.1 0.1 0.1
1 2 3 4
0.2849 0.1279 0.0777 0.0534
0.2849 0.1279 0.0820 0.0587
0.0000 0.0000 5.5606 9.9579
0.6267 0.4677 0.3715 0.3067
0.6267 0.4677 0.3695 0.3034
0.0000 0.0000 0.5474 1.0754
1.8065 1.4444 1.2304 1.0830
1.8065 1.4444 1.2249 1.0726
0.0000 0.0000 0.4481 0.9594
1 1 1 1
1 2 3 4
0.3061 0.1662 0.1291 0.1144
0.3061 0.1662 0.1324 0.1181
0.0000 0.0000 2.5090 3.2096
0.6735 0.5646 0.5144 0.4895
0.6735 0.5646 0.5125 0.4864
0.0000 0.0000 0.3661 0.6290
1.8980 1.6419 1.5296 1.4745
1.8980 1.6419 1.5258 1.4679
0.0000 0.0000 0.2515 0.4455
10 10 10 10
1 2 3 4
0.3902 0.3056 0.2976 0.2965
0.3902 0.3056 0.2986 0.2975
0.0000 0.0000 0.3238 0.3153
0.8585 0.8440 0.8417 0.8413
0.8585 0.8440 0.8415 0.8411
0.0000 0.0000 0.0282 0.0336
2.2320 2.1748 2.1661 2.1648
2.2320 2.1748 2.1654 2.1639
0.0000 0.0000 0.0323 0.0395
220
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
Table 5 Different settings of parameters for Example 2 (Model B) No
l
s
a
b
m1
m2
p1
p2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.5 3 3 3 3 0.5 3 3 3 3 0.5 3 3 3 3
1 1 1 1 1 10 10 10 10 10 1 1 1 1 1
(0.5,1,0.5) (0.5,1,0.5) (5,5,5) (0.5,1,0.5) (0.5,1,0.5) (0.5,1,0.5) (0.5,1,0.5) (5,5,5) (0.5,1,0.5) (0.5,1,0.5) (0.5,1,0.5) (0.5,1,0.5) (5,5,5) (0.5,1,0.5) (0.5,1,0.5)
(0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (0.5,0.5,0.5) (4,4,4) (4,4,4) (4,4,4) (4,4,4) (4,4,4)
(1,1,1) (1,1,1) (1,1,1) (1,10,1) (10,1,10) (1,1,1) (1,1,1) (1,1,1) (1,10,1) (10,1,10) (1,1,1) (1,1,1) (1,1,1) (1,10,1) (10,1,10)
(1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1)
(0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2) (0.2,0.2)
(0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1) (0.2,0.1)
Exp. Ocp. Ch.
It.
Table 6 Exact and approximate results under No. 1 –5 parameter settings in Table 5 for Example 2 (Model B) No
1
2
3
4
5
n
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Loss Handover-1
Loss Handover-2
Loss Fresh Call
Exa.
App.
Exa.
App.
Exa.
App.
Exa.
App.
0.2809 0.1305 0.3598 0.5892 0.2025 0.4036 0.5987 0.2585 0.4958 0.6043 0.0447 0.4739 0.1935 0.3898 0.2588
0.2805 0.1303 0.3605 0.5891 0.2026 0.4050 0.5987 0.2585 0.4962 0.6045 0.0453 0.4809 0.1934 0.3894 0.2575
0.2608 0.1218 0.3718 0.5914 0.2170 0.4221 0.6075 0.2914 0.5168 0.5970 0.0434 0.4700 0.1877 0.3606 0.2600
0.2637 0.1218 0.3717 0.5922 0.2167 0.4218 0.6076 0.2910 0.5166 0.5967 0.0434 0.4670 0.1942 0.3594 0.2600
0.7726 0.5516 0.8273 0.9252 0.6497 0.8547 0.9591 0.8291 0.9459 0.9307 0.2923 0.8755 0.6456 0.7742 0.7216
0.7720 0.5518 0.8273 0.9252 0.6499 0.8544 0.9591 0.8291 0.9458 0.9308 0.2935 0.8735 0.6438 0.7749 0.7217
1.0535 1.5666 1.1991 1.5144 1.7941 1.2769 1.5578 2.1010 1.4627 1.5350 1.0280 1.3454 0.8390 2.1122 0.9816
1.0525 1.5669 1.1990 1.5143 1.7944 1.2762 1.5578 2.1010 1.4625 1.5353 1.0306 1.3405 0.8372 2.1135 0.9818
5
5
5
5
4
Table 7 Exact and approximate results under No. 6 –10 parameter settings in Table 5 for Example 2 (Model B) No
6
7
8
9
10
n
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Loss Handover-1
Loss Handover-2
LossFresh Call
Exp. Ocp. Ch.
It.
Exa.
App.
Exa.
App.
Exa.
App.
Exa.
App.
0.3669 0.2676 0.8319 0.6206 0.3259 0.8340 0.6297 0.3811 0.8408 0.6391 0.1851 0.8568 0.2652 0.4666 0.8269
0.3666 0.2675 0.8319 0.6205 0.3260 0.8342 0.6297 0.3813 0.8409 0.6392 0.1852 0.8575 0.2648 0.4662 0.8266
0.3406 0.2458 0.8274 0.6172 0.3236 0.8313 0.6307 0.3850 0.8386 0.6269 0.1715 0.8328 0.2603 0.4184 0.8218
0.3415 0.2458 0.8274 0.6176 0.3236 0.8312 0.6309 0.3850 0.8386 0.6268 0.1714 0.8328 0.2620 0.4186 0.8218
0.8264 0.6863 0.9861 0.9360 0.7486 0.9867 0.9639 0.8692 0.9908 0.9423 0.5473 0.9868 0.7194 0.8128 0.9849
0.8261 0.6863 0.9861 0.9360 0.7486 0.9867 0.9639 0.8692 0.9908 0.9423 0.5475 0.9868 0.7186 0.8130 0.9849
1.1933 1.8830 1.8135 1.5566 2.0395 1.8179 1.5936 2.2546 1.8294 1.5814 1.5955 1.8196 0.9845 2.2182 1.8067
1.1927 1.8831 1.8135 1.5565 2.0395 1.8179 1.5935 2.2545 1.8294 1.5815 1.5958 1.8196 0.9834 2.2188 1.8067
4
5
5
5
4
A.S. Alfa, B. Liu / Computer Communications 27 (2004) 208–221
221
Table 8 Exact and approximate results under No. 11 –15 parameter settings in Table 5 for Example 2 (Model B) No
11
12
13
14
15
n
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Loss Handover-1
Loss Handover-2
LossFresh Call
Exa.
App.
Exa.
App.
Exa.
App.
Exa.
App.
0.4416 0.2299 0.3670 0.6447 0.2853 0.4089 0.6324 0.2887 0.4736 0.6685 0.1771 0.4605 0.3714 0.4395 0.3244
0.4413 0.2294 0.3669 0.6446 0.2854 0.4098 0.6324 0.2888 0.4740 0.6685 0.1771 0.4661 0.3708 0.4386 0.3216
0.4113 0.1880 0.3608 0.6416 0.2777 0.4144 0.6363 0.3036 0.4851 0.6525 0.1529 0.4249 0.3673 0.3625 0.3178
0.4120 0.1880 0.3608 0.6418 0.2779 0.4142 0.6363 0.3036 0.4850 0.6525 0.1529 0.4235 0.3687 0.3623 0.3180
0.9241 0.7899 0.9061 0.9628 0.8226 0.9179 0.9732 0.8830 0.9573 0.9669 0.7290 0.9190 0.8885 0.8578 0.8850
0.9240 0.7899 0.9061 0.9628 0.8226 0.9178 0.9732 0.8830 0.9572 0.9669 0.7293 0.9187 0.8883 0.8581 0.8850
1.3657 1.9875 1.2669 1.6074 2.1014 1.3323 1.6056 2.1867 1.4423 1.6354 1.8631 1.3439 1.2598 2.2370 1.2028
1.3653 1.9875 1.2670 1.6074 2.1015 1.3321 1.6056 2.1867 1.4422 1.6355 1.8638 1.3422 1.2591 2.2379 1.2029
† It. V the number of algorithm iterations required for convergence. In these tables, we always use n to indicate the label number of nodes. Besides, for simplicity we use notations m ¼ ðm1 ; m2 ; m3 ; m4 Þ; p ¼ ðp1 ; p2 ; p3 Þ; a ¼ ða1 ; a2 ; a3 ; a4 Þ in Tables 1– 4, and a ¼ ða1 ; a2 ; a3 Þ; b ¼ ðb1 ; b2 ; b3 Þ; m1 ¼ ðm1;1 ; m1;2 ; m1;3 Þ; m2 ¼ ðm2;1 ; m2;2 ; m2;3 Þ; p1 ¼ ðp1;1 ; p1;2 Þ; p2 ¼ ðp2;2 ; p2;3 Þ in Tables 5 – 8. (Remark: the definitions of other mathematical symbols, e.g. l; mi and ai ; can be found in Section 2). Let us take Table 1 as a demonstration in reading our numerical results. Under the setting of l ¼ 3; m ¼ ð1; 1; 1; 1Þ; a ¼ ð0:8; 1; 0:5; 1Þ and p ¼ ð0:1; 0:3; 0:2Þ; Table 1 shows that at node n ¼ 3; the loss probability of handover calls is 0.1045 (exact result) and 0.1066 (approximate result) with relative error 2.0337%, and the expected number of occupied channels is 1.4105 (exact result) and 1.4083 (approximate result) with relative error 0.1555%. It has been reported in Tables 1 –4 that the relative errors of our approximate results are smaller than 2.5% for most cases. Even for the worst case (rarely happened), the maximum relative error does not surpass 10%. As a whole, the proposed method appears to produce fairly accurate results. Besides, the number of iterations needed for convergence is moderate, only 4 or 5 (Tables 6 – 8), which shows that our method is time-saving.
5. Conclusion This paper presents an approximation approach to evaluate performances of mobile communication networks with N-cells in tandem. The approximation method is simple and applicable. Two different models are studied, which correspond a uni-directional traffic flow and
Exp. Ocp. Ch.
It.
4
4
4
4
4
bi-directional traffic flows, respectively. Numerical experiment results show excellent performances of our approximation approach for these two models. References [1] R.J. Boucherie, M. Mandjes, Estimation of performance measures for product form cellular mobile communications networks, Telecommun. Syst. 10 (1998) 321 –354. [2] R.J. Boucherie, N.M. Van Dijk, On a queueing network model for cellular mobile telecommunications networks, Oper. Res. 48 (1) (2000) 38–49. [3] A. Brandwajn, Y.L. Jow, An approximation method for tandem queues with blocking, Oper. Res. 36 (1) (1988) 73 –83. [4] D. Everitt, N.W. Macfadyen, Analysis of multi-cellular mobile radiotelephone systems with loss, Br. Telecom Tech. J. 1 (1983) 218–222. [5] D. Everitt, D. Manfiled, Performance analysis of cellular mobile communication systems with dynamic channel assignment, IEEE J. Selected Areas Commun. 7 (1989) 1172–1180. [6] D. Everitt, Traffic engineering of the radio interface for cellular mobile networks, Proc. IEEE 82 (1994) 1371–1382. [7] K.K. Leung, W.A. Massey, W. Whitt, Traffic models for wireless communication networks, IEEE J. Selected Areas Commun. 12 (8) (1994) 1353– 1364. [8] W.A. Massey, W. Whitt, A stochastic model to capture space and time dynamics in wireless communication systems, Prob. Engng Inform. Sci. 8 (1994) 541– 569. [9] D. McMillan, Traffic modeling and analysis for cellular mobile networks, Proc. 13th Int. Teletraffic Cong. (1991) 627–632. [10] D.L. Pallant, P.G. Taylor, Approximations of performance measures in cellular mobile networks with dynamic channel allocation, Telecommun. Syst. 3 (1994) 163–319. [11] D.L. Pallant, P.G. Taylor, Modelling handovers in cellular mobile networks with dynamic channel allocation, Oper. Res. 43 (1) (1995) 33–42. [12] M. Sidi, D. Starobinski, New call blocking versus handoff blocking in cellular networks, Wireless Netw. 3 (1997) 15 –27. [13] W. Li, A.S. Alfa, A PCS network with correlated arrival process and splitted-rating channels, IEEE J. Selected Areas Commun. 17 (7) (1999) 1318– 1325. [14] W. Li, A.S. Alfa, Channel reservation for handoff calls in a PCS network, IEEE Trans. Vehicular Technol. 49 (1) (1999) 1318–1325.