A resource allocation scheme for capacity improvement in the downlink of wideband CDMA networks

Computer Communications 27 (2004) 1295–1313 www.elsevier.com/locate/comcom

A resource allocation scheme for capacity improvement in the downlink of wideband CDMA networksq Oladipupo O. Oyefuga, Mustafa K. Gurcan*, Zenon Ioannou Department of Electrical and Electronic Engineering, Communications and Signal Processing Research Group, Imperial College, Exhibition Road, London SW7 2BT, UK Available online 12 March 2004

Abstract This paper shows the existence of a limit on the maximum throughput that can be achieved in a W-CDMA cell. This limit is due to cochannel interference and is dependent on the number of users transmitting simultaneously. A probability transmission scheme is suggested to increase the throughput limit by reducing the number of users transmitting together. The probability transmission scheme is implemented by matching the arrival process to discrete transmission rates and probabilities. To combat wastage of resources, a threshold based buffer management scheme is used in conjunction with the probability transmission scheme. Simulations show that over 35% capacity improvement is achievable while keeping the average packet delay within tolerable limits. q 2004 Elsevier B.V. All rights reserved. Keywords: Wideband CDMA; Resource allocation; Buffer management scheme; Green’s function

1. Introduction Third generation (3G) mobile networks will deliver high data-rate multimedia applications in a mobile environment [9,19]. These networks are based on Wideband CDMA (W-CDMA) as the air interface technology, linking the base station to the mobile station. A key problem in system design is ensuring the transmission of data to all users, within the required Quality of Service (QoS) parameters. The major QoS parameters of interest are the Bit Error Rate (BER) and Average Packet Delay of the transmitted data. W-CDMA provides a packet switching capability by allocating system resources to individual transmissions. These resources consist of transmission rates obtained using channelisation codes, and transmission power [9,19]. Various solutions have been developed for ensuring QoS targets are met during the resource allocation process [1 – 5]. Bambos et al. [1] examine a distributed allocation of resources to meet BER requirements. They note that when a user is admitted to the network, the allocated transmission q This paper is part of the Special Issue (International Conference on Networks (ICON 2002)—Guest Editors: G.-S. Poo, L. Zhang and L.-H. Ngoh) which was published as COMCOM 27/2, 2004. Guest Editorial doi: 10.1016/S0140-3664(03)00209-3. * Corresponding author. Tel.: þ44-20-7594-6264; fax: þ 44-20-75946302. E-mail address: [email protected] (M.K. Gurcan).

0140-3664/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2004.02.008

power to existing users must be increased to achieve the same BER target. The same situation also arises when the transmitted bit-rate to each user is increased. This is due to the increased channel interference generated by the newly admitted user and is characteristic of CDMA systems, where all transmissions share the same bandwidth but are separated by orthogonal channelisation codes. The channel interference is due to multipath propagation distorting the orthogonality of the codes and was identified by Viterbi in Refs. [2,10]. Multipath propagation causes the downlink transmissions received at each mobile terminal to no longer be orthogonal, and implies that transmissions to any mobile station contribute to the interference experienced by all other mobile stations. As the transmitted bitrate to a set of users is increased, the solution for the total transmission power given in Ref. [1] tends to infinity. In this paper, we show that even if an infinite amount of transmission power were available, there is a limit to the total transmission rate that can be supported in a W-CDMA cell. This limit is due to the increased interference generated as the transmitted bit-rate increases, and is partly dependent on the number of users in the cell transmitting simultaneously, Nav : In practical systems only a finite amount of resources are available for allocation, therefore efficient resource allocation schemes are desirable to make optimal use of these limited resources. This allocation problem is accentuated on

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the downlink due to the asymmetric nature of multimedia traffic, where a greater load is expected. Maximising the downlink throughput within these limited resources is also an appealing objective, as the network operator’s revenue is often dependent on throughput. This objective is explored in Refs. [7,11,12], but such methods are still limited by the same capacity upper bound given in this paper. A possible way of overcoming the capacity limit is to reduce the amount of interference in the cell. We show that this can be achieved by decreasing the number of simultaneous transmitting users in the cell. Berggren et al. [8] suggest the use of one by one scheduling, where the base station transmits to only one user at a time. This method has the advantage of eliminating co-channel interference. However, the system operates in TDMA representing a waste of capacity. The situation may arise that during a user’s allotted time the user requires less capacity than is available. In this paper, we take the approach of allowing an average number of users ðNav Þ less than the number of users in the cell ðNÞ to transmit simultaneously. This ensures that the amount of interference encountered by each transmission is reduced and the system capacity increased. Kim and Honig [4] consider separating users into service classes and allocating bit-rate and channelisation codes to individual users to control the average packet delay, whilst maximising capacity. Power is allocated such that not all BER targets are met in a given frame, but that delay targets are met for all users. However, all transmissions occur simultaneously, increasing interference and limiting the system capacity. The resource allocation scheme proposed in this paper reduces the number of simultaneous downlink transmissions by allowing the base station transmit to each user with a probability Pr , 1; rather than simultaneously to all users. This implies that the average number of users the base station transmits to in each frame, Nav ¼ Pr N: This reduces the interference and increases the capacity. With this scheme, the interval between successive transmissions to each user is increased. This may lead to a large value for the average packet delay unless suitably high transmissions rates are allocated during transmission instances. To minimise wastage in the allocation process, the allocated transmission rates must match the user’s demand. This may also be seen as matching the arrival rate of packets to the base station, to the transmission rate from the base station to the user’s mobile [6]. Assuming packets arrive at the base station as a Poisson process with average arrival rate, l; then

l¼

X X

pj Rj :

ð1Þ

j¼1

Rj is the transmission rate of each code j allocated to the user, pj is the probability that code j is available for transmission and X is the total number of codes allocated. The values of Rj and pj are chosen large enough so that the user’s buffer is cleared of all packets at a single

transmission. The base station will not attempt to transmit to that user again till enough packets are accumulated. Using this, the base station transmits to each user with an average probability Pr : This paper introduces a rate matching method of determining suitable values of Rj and pj ; by matching the arrival rate of packets to a set of discrete transmission rates and probabilities. Together with a threshold based buffer management scheme [13], which uses the number of queued packets to indicate when and which transmission rate and probability pair to use, the ratematching scheme ensures that a transmission probability of Pr is achieved. The probability transmission scheme results in bulk transmissions, which could lead to an increase in packet delay. To ensure that the Average Packet Delay QoS target is satisfied, the stochastic behaviour of the threshold based buffer management scheme must be known. Analyses for similar threshold based buffer management schemes exist in Refs. [17,18]. However, these do not meet the requirements of the buffer management scheme proposed in this paper and shown in Fig. 9, as the authors in Refs. [17,18] do not give the analysis for a bulk arrival—bulk departure discrete time threshold controlled queue. In this paper we show that the delay performance of the buffer management scheme can be analysed using probability-generating functions (pgf). For systems such as the proposed threshold buffer management scheme, determining the pgf can be difficult. One method is to determine the pgf of finite length buffers and then combine them to produce the pgf of the entire threshold based buffer. This paper extends the Green’s function principles used in Ref. [17], to determine the pgf of finite length buffers, from the easily derived pgf of infinite length buffers. The pgfs of the finite length buffers are then combined to achieve the pgf of the threshold based buffer management scheme. Using the methods discussed above, graphs for the total transmission rate versus power and the average packet delay versus total transmission rate are determined for both the simultaneous transmission scheme derived from equations presented in Ref. [1] and the probability transmission scheme discussed in this paper. We show that over 35% capacity improvement in total transmission rate is achievable while keeping the Average Packet Delay within tolerable bounds. The rest of this paper is organised as follows: Section 2 discusses the system model and air interface model considered and details the limits on system capacity. Section 3 presents the resource allocation scheme, explaining the optimal determination of code allocation probabilities to each user, such that codes are available to match the arrival rate of packets. Section 4 considers the integration of resource allocation into a threshold buffer management scheme and its use to determine that the Average Packet Delay QoS target is achieved. Results are presented and discussed in Section 5. Section 6 details our conclusions.

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2. Wideband CDMA system model The downlink of the 3G system cell considered in this paper is depicted in Fig. 1. Data packets arrive from a variety of public networks, either packet switched (Internet, ISDN, etc.) or circuit switched (PSTN) and are routed over the mobile operators network, arriving at the base station of the desired user’s cell. It is assumed that packets arrive according to a known arrival process and are buffered until resources are allocated for their onward transmission to the mobile user. The base station allocates the available downlink resources to ensure that each user’s QoS demands are met. Each transmission is allocated an Orthogonal Variable Spreading Factor (OVSF) code, determining the transmission bit-rate, ri ; and associated transmission power, Pi : The total transmission power allocated to all N users in the cell is given as PT ¼

N X

Pi :

ð2Þ

i¼1

A simplified model for the resource allocation process between the base station and the mobile user is shown in Fig. 2. In order to examine the fundamental limits, which restrict the bit-rate capacity of the link, we firstly require an interference model to investigate the interference generated by each transmission. This can be used in a power allocation model to determine how power levels can be optimally set for each user. An analysis of this model will demonstrate the source of an upper bound on performance. 2.1. Interference model The base station allocates transmission rates and power to ensure that the QoS targets of the downlink transmissions

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are achieved. However, since all transmissions occur simultaneously, the downlink transmission to each user is seen as co-channel interference at all other users’ receivers. Viterbi [10] suggests that even when the channelisation codes used are orthogonal, multipath propagation on the channel may lead to a loss of orthogonality between codes, providing a source of co-channel interference. Berggren et al. [8] amongst others, model this loss of orthogonality by a factor denoted as a: This factor represents the proportion of the power transmitted to other users in the downlink, received as non-orthogonal. The channel model used incorporates the transmitter and detector and has a fixed chip rate, W. The ith user’s channel path gain is represented by hi : Three sources of interference are identified: 1. Normal channel noise N0 2. Interference generated by downlink transmission in the surrounding cells, Iinter;i 3. Intra-cell or co-channel interference from power not used for transmissions to the ith user, modeled by orthogonality factor, a This interference is shown in Fig. 3 below. 2.2. Power allocation model Transmission power is allocated in order to satisfy the BER QoS target of all transmissions in the W-CDMA system. The BER is a function of the Eb =N0 (Energy Utilisation Efficiency) at the output of the user’s receiver. For the ith user, the Eb =N0 can be modeled as a function of the power allocated to the user’s transmission; the spreading factor of the code allocated for the transmission (itself a function of ri ; the transmission rate allocated to the ith user); and the interference present on the channel.

Fig. 1. 3G network diagram.

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Fig. 2. Simplified diagram of W-CDMA resource allocation.

Therefore, in order to achieve the BER target, the transmission power allocated to each user, Pi ; is a function of the allocated bit-rate ri ; and the channel and other interference parameters. To avoid any wastage of power, we assume that the bit error rate is fixed at the maximum that the QoS requirements allow. This implies a target Eb =N0 for each user, defined as Gi : ðEb =N0 Þi ¼ Gi :

ð3Þ

ðEb =N0 Þi can be expressed in terms of the spreading factor and SNIR at the input to the user receiver, based on the interference model described by Fig. 3. ðEb =N0 Þi ¼ SFi £ SNIRi ¼

W Pi h i X ; ri a Pj hi þ N0 þ Iinter;i j–i

ð4Þ

Substituting the constraint for ðEb =N0 Þi given in Eq. (3), this expression can be re-arranged into a matrix form as suggested in Ref. [1] X j–i

Pj

ðN0 þ Iinter;i ÞGi ri aGi ri ¼ ; W Whi

ð5Þ

ðI 2 FÞ·P ¼ u;

8 > < 0; Fij ¼ aGi ri > : W

9 if i ¼ j > = : ; if i – j >

Re-arranging Eq. (6), and assuming the solution exists as stated by Bambos et al. [1], the solution is simply Eq. (9), satisfying the target Eb =N0 in Eq. (2). PðI 2 FÞ21 u:

ð9Þ

An iterative solution for this can be found using Distributed Power Control [1].

Eq. (9) is a general solution for the W-CDMA model, based on the interference model presented in Section 2.1 and will form the basis for the rest of this paper. The authors in Ref. [1] show that for a fixed BER, as the number of users in the system N is increased, and given that a solution exists, PT the total transmission power required increases towards infinity. Alternatively, for a fixed number of users, PT will also increase as the transmission bit-rate to each user is increased. We show that there exits a finite total bit-rate for which PT is infinite. This implies that even with unlimited transmission power, there is a limit to the total bit-rate that

ð6Þ

where P ¼ ½P1 P2 · · ·PN T is the column vector of individual transmission powers u¼



ðN0 þ Iinter;1 ÞG1 r1 ðN þ Iinter;N ÞGN rN ; · · ·; 0 Wh1 WhN

T

ð8Þ

2.3. Constraint on maximum system throughput

i; j ¼ 1; 2; …; N:

Pi 2

and F is a non-negative N by N square matrix, with entries

ð7Þ Fig. 3. Interference model for ith user.

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can be delivered for a given BER QoS target. This limit exists and can be derived. The solution given by Eq. (9) must satisfy the expression for PT given in Eq. (1). From Eq. (1) N X

Pj ¼ PT 2 P i :

ð10Þ

j–i

Substituting for Eqs. (3) and (10) into Eq. (4) and rearranging, the target Eb =N0 for the ith user can be expressed as

Gi ¼

W ri

Pi

aðPT 2 Pi Þ þ

N0 þ Iinter;i hi

:

ð11Þ

We assume the special case where all users have the same BER targets and are allocated equal transmission rates. For simplicity of analysis, Iinter;i is fixed for all users and does not vary with PT ; however, the analysis is valid when Iinter;i varies with the user’s position and PT [1]. In this case, the Eb =N0 target is defined as G and the allocated transmission rate as R: In the limit, as PT tends to infinity, the following expression results. lim G ¼

PT !1

W Pi : R aðPT 2 Pi Þ

ð12Þ

With infinite power available, the power allocated to each user is identical and can be expressed simply by Pi ¼

PT : N

ð13Þ

Substituting for Eq. (13) in Eq. (12) and rearranging, the bit-rate allocated to each user is R¼

W : GaðN 2 1Þ

ð14Þ

For all N users in the system, the total bit-rate that can be delivered RT;max ; is given as RT;max ¼ RN ¼

WN : GaðN 2 1Þ

ð15Þ

Eq. (15) implies that even for an infinite amount of transmission power, the total bit-rate that can be delivered by the system has a finite upper bound, which is a function of the number of simultaneously transmitting users in the system N: From Eq. (11), it can be seen that as PT increases, the intra-cell interference becomes the dominant interference term and is the limiting factor on the system performance. Fig. 3 shows that this term is a function of the transmission power of all the other users in the system. As the transmission rate of each user is increased, more power is required to achieve the BER target. However, the increased power allocated to that user is seen as increased interference by all other users. To overcome this added interference, other users in turn will require more power to satisfy their BER targets. So, for a fixed BER target, as the bit-rate is

Fig. 4. Average allocated power versus total offered transmission rate.

increased, the required power will increase quickly, tending towards infinity as the total rate approaches RT;max : Simulations, as described in Section 5, were performed showing the total transmission power required against the total delivered bit-rate for a base station with N ¼ 20 users, a ¼ 0:6; G ¼ 4 dB and W ¼ 4:096 Mb/s. All users allocated equal bit-rates. The graph is shown in Fig. 4, where it can be seen that the maximum total bit-rate obtained is about 2.8 Mb/s, and confirmed in Eq. (15). In practical systems, due to equipment and interference limitations, the total transmission power that can be allocated is constrained to a finite value, reducing the total deliverable bit-rate. However, by increasing the limit, the bit-rate that can be delivered for finite powers will also be increased. Fig. 5 shows how RT;max ; the maximum throughput that can be supported by the system, varies as a function of the number of users present in the system, for different orthogonality factors. Both Eq. (15) and Fig. 5 imply that if the average number of users transmitting simultaneously is reduced, the maximum total deliverable bit-rate that can be supported in the system in a single frame will be increased. We propose that in a given frame, the base station transmits to each user with probability Pr ; where Pr , 1:

Fig. 5. Maximum throughput limit variations with N.

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This reduces the average number of users the base station transmits to simultaneously ðNav Þ; and thereby increasing RT;max : With this method, Nav becomes Nav ¼ Pr N:

ð16Þ

As Pr determines the average number of users simultaneously transmitting, it is necessary to determine the value of Pr that maximises the total throughput over the channel.

station will then not transmit to that user again till enough packets are accumulated. If the bit-rates are chosen appropriately, it is possible to ensure that each user only gets transmitted to with probability Pr on average. In Section 3, we present a method that matches the allocation of discrete transmission rates to the arrival process of a user.

3. Arrival rate to departure rate matching 2.4. Optimal channel access probability In this section we determine the value of Pr that maximises the total throughput that can be achieved in the channel. Assuming that the base station transmits to each user with probability Pr ; the probability of the base station transmitting simultaneously to a group of M users out of the total N users in the cell can be expressed as a binomial combination as ! N N2M PðM from NÞ ¼ PM : ð17Þ r ð1 2 Pr Þ M The average system throughput Savg ; can then be described as this probability multiplied by the maximum throughput that can be achieved by that group of M users, RTmax;M ; given in Eq. (15). ( ) ! N X N M N2M RTmax;M : Pr ð1 2 Pr Þ ð18Þ Savg ¼ M M¼1 The value of Pr that maximises the expression in Eq. (18) is the probability for which dSavg =dPr is equal to zero. This can be computed from Eq. (19) using the Newton – Raphson method (0 1 N N X dSavg @ A{MPðM21Þ ¼ ð1 2 Pr ÞðN2MÞ 2 ðN 2 MÞ r dPr M¼1 M ) N2M21  PM } RTmax;M ¼ 0: r ð1 2 Pr Þ

ð19Þ

Though Eq. (19) provides the transmission probability that maximises the system throughput, it does not take account of the arrival process of packets to the base station buffers. Situations may arise where the buffers are empty or not enough packets are queued in the buffer when the base station tries to transmit to the user. To avoid resource wastage, it would be best to match the allocated transmission rates to the users demand, that is the user’s arrival rate of packets. It is required that each user’s transmission is only allowed to access the channel every 1=Pr frames on average. To avoid excess packet delay, the allocated rates to each user must be able to cope with the packets that accumulate during periods of no transmissions. The allocated bit-rates must be such that most of the accumulated packets are transmitted during one transmission instance. The base

In each frame, the base station will allocate each user’s buffer a channelisation code of rate Rj and will try to transmit to that user with probability pj : In this section Rj is the transmission rate in packets/frame, where 1 packet/frame is equivalent to a bit-rate of 16 kb/s. A W-CDMA frame is 10 ms long and there are 160 bits per packet. The values for Rj and pj are matched to the arrival rate of packets to the base station buffers. The allocation process is such that the allocated rates can cope with the arrival rates of packets when the base station transmits to each user with probability Pr : 3.1. Bit-rate allocation model In W-CDMA systems, the base station transmits to users using the entire available bandwidth, W: Channelisation or OVSF codes are used to separate the different simultaneous transmissions [9,19]. OVSF codes are constructed in a tree like manner shown in Fig. 6. The OVSF code tree structure consists of several levels of codes, with each level containing codes of the same length and transmission rate. The channel separation (or orthogonality) between two codes is achieved if there is no direct path between them. For example, in Fig. 6, codes 1 and 2 are not orthogonal and cannot be jointly assigned, but codes 1 and 3, or codes 2 and 3, are and can be jointly assigned. Since codes at each level are of different lengths, different level codes provide different transmission rates and allow W-CDMA systems to provide a packet switching ability [1]. 3.2. Allocation of transmission rates and probabilities Packets arrive for each user at the base station; with average arrival rate l measured in packets/frame, and are then buffered for transmission. To ensure optimal use of resources, we wish to match the transmission rate of packets to the arrival rate. For simplicity, we assume packets arrive based on a Poisson process, the probability of j packets arriving for a user in one frame is given by PðjÞ as PðjÞ ¼

ðlj Þe2l : j!

ð20Þ

The arrival process need not be Poisson, but it is essential that its probability density function be known. Fig. 7 shows PðjÞ as a function l for different values of j:

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Fig. 6. OVSF code tree.

A unique way of ensuring that the transmission rate is exactly matched to the arrival rate is to set Rj ¼ j and pj ¼ PðjÞ from Eq. (20) for any given value of l: Using the definition of the average arrival rate X X l¼ jPðjÞ ¼ p j Rj : ð21Þ j

j

Eq. (21) implies that given any value of l; the arrival rate can be matched to a set of discrete transmission rates and probabilities using Eq. (20) and Fig. 7. If l ¼ 1; using the values of j and PðjÞ at l ¼ 1 in Fig. 7, the set of discrete rates and probabilities can be obtained as (R0 ¼ 0, p0 < 0.39), (R1 ¼ 1, p1 < 0.38), (R2 ¼ 2; p2 < 0:19), (R3 ¼ 3; p3 < 0:07), (R4 ¼ 4; p4 < 0:02). The base station then has the choice of transmitting to the user with rate Rx at probability px ; where x ¼ 1; …; 4: However, to increase the system capacity, transmissions to each user must occur with probability Pr ð0 , Pr , 1Þ: The average interval between successive transmissions to each user is then given as 1=Pr : To ensure the allocated rates can cope with the accumulated packets from the periods of no transmission, the rates and probability in Eq. (21) must be adjusted to take account of the probability transmission scheme. Observing only at the instances of transmission, the effective arrival rate for a single user would be

leff ¼

l : Pr

p3 R3 þ p4 R4 , ðp3 þ p4 ÞR4 :

ð24Þ

ð22Þ

Allocated rates and probabilities that match the arrival rate and can cope with the accumulation of packets are then obtained by examining Fig. 7 at leff and not at l as stated earlier. Thus, Eq. (22) becomes X leff ¼ p j Rj : ð23Þ j

It should be noted that though the rates allocated for leff in Eq. (23) are greater those at l in Eq. (21), the base station will only transmits to each user with probability Pr when using the rates in Eq. (23). Fig. 7 assumes that it is possible to allocate rates Rj in unit steps, such that Rj [ {0,1,2,3,4,5,…}. From Section 3.1 and Ref. [19], transmission rates are only available in discrete multiples of 2, i.e. Rj [ {0,1,2,4,8,16,…,256}, implying some rates in Fig. 7 do not exist. To overcome this, if a rate assigned using Fig. 7 does not exist, the next highest transmission rate must be assigned, e.g. since Rj ¼ 3 does not exist, Rj ¼ 4 must be assigned instead. The new probability for the transmission code Rj ¼ 4 is obtained by summing p3 and p4 : Thus, instead of the base station using a rate R3 with probability p3 ; the user must be assigned rate R4 with probability ðp3 þ p4 Þ: Fig. 8 shows the modified Fig. 7 using only the allowable rates and the adjusted probabilities as a function of l. If only the allowable transmission rates are used, i.e. Rj [ {0,1,2,4,8,16,…,256}, and since R4 q R3 ;

Fig. 7. Packet arrival probability variation with average arrival rate (j ¼ 0 to 7).

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that the OVSF code is fully utilized during the transmission period, thereby preventing wastage of resources. Applying the principle to the buffer in Fig. 9 and taking nðkÞ as the number of packets in the buffer at the kth time frame, ri as the transmission rate and pi as the transmission probability, the transmission process of the buffer can be described as

Fig. 8. Packet arrival probability variation with average arrival rate.

Eq. (24) implies that arrival rate is no longer equal to the transmission rate, but rather

leff ,

X

p j Rj :

ð25Þ

j

Eq. (25) represents a waste in capacity as more resources than the average arrival of packets are allocated to the user. To avoid this wastage, a threshold based buffer management scheme is proposed to ensure that transmissions only occur when there are enough packets queued in the buffer. This delays transmissions until enough packets are stored in the buffers to ensure that the allocated transmission rates are fully utilized during the transmission instance.

4. Threshold based buffer management scheme The proposed buffer management scheme is shown in Fig. 9 and also discussed in Ref. [13]. The buffer management scheme operates as follows: packets arrive from the external networks with an average arrival rate l: The packets are queued in the base station buffers, which are assumed to be of infinite length, and have a number of thresholds along its length. When the number of queued packets crosses a threshold Ti ; the base station attempts to transmit packets from that buffer with rate RTi at a probability pTi RTi and pTi represents the allocation of codes with non-zero probabilities at the desired values of leff as obtained in Section 3. The thresholds are set such that RTi ¼ Ti ; so that transmissions at rate RTi cannot occur until at least RTi packets are queued in the buffer. This ensures

0 # nðkÞ , T1

ri ¼ 0

pi ¼ 0

T1 # nðkÞ , T2

ri ¼ RT1

pi ¼ pT1 :

T2 # nðkÞ

ri ¼ RT2

pi ¼ pT2

ð26Þ

It should be noted that based on Eq. (25) in Section 3.1 and Fig. 8, the values of pT1 and pT2 changes as l varies. In Fig. 8, it is seen that only a few of the transmission rates Rj have probabilities that are significantly greater than zero for any value of leff ; especially if Pr is low enough. Only transmission rates with non-zero values are assigned, so the threshold buffer management considers the case where only two or three codes are assigned. As buffers must wait until enough packets are available before transmission, some packets may remain un-transmitted for several time frames. It is essential that the average packet delay QoS target be not violated in the proposed buffer management scheme. To ensure this, an analytical method must be established for predicting the average packet delay that is achieved in the buffer management scheme for a given average arrival rate, transmission rate and probability allocation and threshold settings. Simple queuing analysis assumes independence between the arrival process, queue length and server rates and cannot be used to analyse this buffer management scheme. Ibe and Keilson [16] examine the continuous time case of a similar threshold based buffer scheme. The authors assume that packets arrive singly with exponentially distributed interarrival times. Lui and Golubchik [17] state that adapting the method in Ref. [16] to bulk arrival processes is difficult and introduce a bulk arrival continuous time analysis for a similar system. Refs. [16,17] deal with continuous time situations and are not directly applicable to the scheme in this paper. We extend the principle of Green’s function method used in Ref. [16], to analyse the buffer management scheme as a discrete time bulk arrival queuing process.

Fig. 9. Threshold based buffer management scheme.

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Fig. 10. Simple Markov process (Chain A).

4.1. Analysis of average packet delay of threshold based buffer management scheme One suitable method of analysing the stochastic behaviour of a threshold based buffer management scheme is to use probability generating functions (pgf) [15]. Determining the pgf of the queuing process in Fig. 9 is difficult. Fig. 9 may be seen as composed of two finite buffers (the first from 0 to T1 2 1 and the second from T1 to T2 2 1) and an infinite buffer from T2 to infinity, with different rates and probabilities acting in each of these buffers. The Green’s function methods allows us to analyse each buffer separately and then combine the separate analyses to obtain the pgf of the entire process in Fig. 9. Using the Green’s function method, each of the finite buffers is analysed as in the following steps: 1. Given any buffer, first analyse an infinite Markov chain having a state space stretching from 21 to þ1: This infinite chain must have the same arrival and departure process as the desired buffer. The equation governing this infinite chain is known as the ergodic Green’s function. 2. The transition equations for the desired finite buffer are obtained using the arrival and departure processes, which are assumed known and non-time varying. 3. The transition differences between the infinite chain in step 1 and the desired finite buffer in step 2 are obtained in terms of the unknown probabilities of the states of the finite buffer. These equations are called the compensating functions. 4. The compensating functions obtained in step 3 are then convolved with the ergodic Green’s function derived in step 1 to produce a set of T equations (where T is

the length of the finite buffer) in terms of the unknown probabilities and the known ergodic Green’s function. 5. The simultaneous equations are solved to obtain the unknown probabilities of the states of the finite buffer. A simple illustration of the Green’s function approach is illustrated below. Assume a simple discrete Markov chain (Chain A) representing a queuing process shown in Fig. 10. For simplicity, the state space of Chain A ðNk Þ; is defined on the set of positive integers, i.e. 0 # Nk # 1: As both the arrival and departure processes are assumed known and non-time varying, the transition probability from state n 2 x to state n depends on the arrival and departure processes and the difference between the states. This transition probability is defined as ax and can be determined as shown in Appendix A. Using steps 1 – 5 above, the pgf of Chain A is determined as follows. Step 1 First, represent Chain A by an infinite Markov chain called Chain B with its state space stretching from 21 to 1, and determine the ergodic Green’s function of Chain B (Fig. 11). Chain B must have the same arrival rates and departure process as Chain A. However, as the state space for the Chains A and B are different, their transition probabilities are slightly different. The transition probability for Chain B will be denoted as aH x : The ergodic transition equation for Chain B can written as in Ref. [16] as gn ðkÞ ¼ P½Nk ¼ n=N0 ¼ 0:

ð27Þ

gn ðkÞ is defined as the probability of Chain B being in state n at the kth time interval, given that the system started in state 0 when k ¼ 0: Eq. (27) can be written as a one-step

Fig. 11. Newly defined infinite Markov process (Chain B).

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O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

transition equation 1 X

gn ðk þ 1Þ ¼

as

gn2x ðkÞaH x:

ð28Þ

pj ¼

x¼21

m!1

As the term gn ðkÞ in Eq. (27) decays exponentially with k [18], Eq. (29) has a finite sum as m tends to infinity. The ergodic Green’s function of Eq. (28) is obtained as " #  m m 1   X X X  H  gn ðk þ 1Þ ¼ gn2x ðkÞax  ; ð30Þ   x¼21

k¼0

m!1

1 X

gn 2 gn ð0Þ ¼

k¼0

n¼21

¼

! n2x

gn2x z

1 X

! x aH xz

:

ð32Þ

x¼21

By definition, at P k ¼ 0; gn ðkÞ ¼ 1 when n ¼ 0; and gn ðkÞ ¼ 0 n if n – 0: The term 1 n¼21 gn ð0Þz is equal to 1 when n ¼ 0 and zero for all other values of n. After further simplification, Eq. (32) becomes ð33Þ P1

pi aj2i :

pi aj2i 2

! pi aH j2i

:

ð35Þ

i¼0

i¼0

Steps 4 and 5 Taking the z-transform of both sides of Eq. (35), we obtain ! 1 1 1 1 X X X X j H pj z ¼ pi aj2i zj þ c j zj j¼0

j¼0 1 X

i¼0

j¼0

1 !0 1 X H j2i pi z @ aj2i z A þ CðzÞ: i

i¼0

ð36Þ

j¼0

P The term pj zj is the probability generating function PðzÞ; of Chain A [15]. Simplifying Eq. (36) PðzÞ ¼

CðzÞ ¼ CðzÞGðzÞ: 1 2 AH ðzÞ

ð37Þ

Eq. (37) shows that the pgf of Chain A, PðzÞ; is obtained by the multiplication of the z-transform of the ergodic Green’s function GðzÞ; and the z-tranform of the compensating functions CðzÞ: From the convolution property of z-transforms, the probability pj of any state of Chain A can be obtained as the convolution of the ergodic Green’s function with the compensating functions as pj ¼

1 X

cm gj2m :

ð38Þ

m¼21

GðzÞ is defined as n¼21 gn zn and is the z-transform of the ergodic Green’s function in Eq. (29). AH ðzÞ is the z-transform of the transition probabilities of Chain B. Step 2 Step 2 involves determining the transition equations for Chain A. Since the transition probability between states n and n 2 x is defined as ax ; the transition equation for Chain A can be written as the single step Kolmagorov equation [15] as 1 X

i¼0

ð31Þ

n¼21 x¼21 1 X n¼21

pj ¼

þ

1 X

The compensating function at state j is defined as ! 1 1 X X H cj ¼ pi aj2i 2 pi aj2i :

¼ gn2x aH x:

Taking the z-transform of both sides of Eq. (31) gives " # 1 1 1 X X X n H ðgn 2gn ð0ÞÞz ¼ gn2x ax zn

1 : 12AH ðzÞ

1 X i¼0

m!1

x¼21

GðzÞ ¼

pi aH j2i

i¼0

The ergodic Green’s function for Chain B, which is time independent, is defined as  m  X  gn ¼ gn ðkÞ : ð29Þ  k¼0

1 X

ð34Þ

i¼0

pj is the steady state probability of state j in Chain A, which is unknown. Step 3 The next step is to determine the compensating functions, which show the transition differences between Chains A and B, due to the difference in their state spaces. The compensating functions are obtained by modifying Eq. (34) with the transition probabilities of Chain B. Thus, without changing the overall value, Eq. (34) can be modified

An expression of the form of Eq. (37) is not always obtainable. Assuming Chain A were of length T 2 1; the aim of the analysis by Green’s function method, is to obtain T simultaneous equations in the form of Eq. (38) for the probabilities pj ð0 # j # T 2 1Þ of the finite buffer. These simultaneous equations can then be solved to obtain the probabilities pj : Using the approach above, the analysis for the finite buffers in Fig. 9 is presented in Appendix A.1. The compensating functions are determined in Appendices A.2 and A.3 for both finite buffers. The simultaneous equations for each of the buffers are derived in Appendix A.5. The infinite buffer (from T2 to 1) is approximated by the probability generating function of an infinite buffer served by a rate RT2 with probability pT2 : The analysis for this is shown in Ref. [14] and presented in Appendix B. The three separate analyses for the different buffers are combined using simple queuing theory methods in Appendix A.6 to determine the probabilities P0 ; PT1 ; PT2 ; which represent the probability of the number of packets being between (0 to T1 2 1), (T1 to T2 2 1) or (T2 to 1) respectively.

O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

1305

Assuming a Poisson arrival rate of packets to the buffers, the pgf of the buffer threshold scheme is given in Eq. (A41) as 0 1 ! TX TX 1 21 2 21 n n PðzÞ ¼P0 p0;ðnÞ z þ PT1 @ p1;ðnÞ z A n¼T1

n¼0

{ðRT2 2 lÞ 2 RT2 ð1 2 pT2 Þ}ðz 2 1Þ zRT2 e2lðz21Þ 2 ð1 2 pT2 ÞzRT2 2 pT2 ! RY T2 21 z 2 zR :  1 2 zR R¼1 þ PT2

ð39Þ

The mean number of packets in the buffer is derived by evaluating the first moment of PðzÞ at z ¼ 1 [15], to obtain 0 1 ! TX TX 1 21 2 21 q ¼P0 ð1Þ ¼ P0 np0;ðnÞ þ PT1 @T1 þ np1;ðnÞ A n¼0

n¼T1

2lRT2 2 l2 2 RT2 ðRT2 2 1Þ þ S00 ð1Þ þ PT2 T2 þ 2{ðRT2 2 lÞ 2 RT2 ð1 2 pT2 Þ} ! RT2 21 X 1 þ : ð40Þ 1 2 zR R¼1 S00 ð1Þ is the second moment of Sð1=zÞ given in Appendix B. The values of pT1 and pT2 are determined based on the arrival rate l the choice of Pr from the discussions in Section 3.2. The discrete queuing model assumes that data packets can only enter service at the beginning of each time frame. In practical systems, packet arrivals can occur at any time. An approximate average of l=2 packets arrive during each service interval for Poisson arrivals [14], this has the effect of adding 1/2 time frame to the average packet delay. The mean queue size is given by n ¼ q þ

l : 2

ð41Þ

Using Little’s formula [15], the average packet delay, normalised to units of time frames is given by n d ¼ : l

ð42Þ

Fig. 12 shows the average packet delay as a function of the arrival rate l; for both analysis and simulation. Fig. 12 shows the analytical results conform to those derived via simulations, confirming our ability to predict the average packet delay of the buffer management scheme for any value of l; pT1 and pT2 : In Fig. 12, a special case is assumed where pT1 and pT2 remain constant as l changes. The parameters used are pT1 ¼ p16 ¼ 0:23; pT2 ¼ p32 ¼ 0:77; RT1 ¼ T1 ¼ R16 ¼ 16 packets/frame, RT2 ¼ T2 ¼ R32 ¼ 32 packets/frame. These parameters result from assuming an arrival rate of l ¼ 2; with a desired Pr ¼ 0:1: The peak observed in Fig. 12 at very low arrival rates occurs because at these values of arrival rate, the number of packets in the buffer fluctuates mainly between 0 and T2 : At this point the packets tend to wait longer before being

Fig. 12. Average packet delay versus average arrival rate.

transmitted. As l is increased, the queue frequently exceeds T2; where a greater transmission rate is allocated by the buffer management scheme. This results in a lower average packet delay. However, our main interest is in the peak at l ¼ 2: As long as this peak does not violate the average packet QoS, the average packet delay performance is acceptable. The probability of the buffer transmitting can be calculated using Eq. (43) as Ptransmission ¼ PT1 pT1 þ PT2 pT2 :

ð43Þ

Simulations confirm that by assigning these rates and probabilities obtained via Section 3, the actual transmission probability of the base station to each user is about 0.097, which is close to the desired transmission probability of Pr ¼ 0.1 and has an average packet delay of about 8.5 frames. Using the analysis presented in the previous sections, the system throughput and average packet delay that can be achieved using probability transmission with the buffer threshold scheme, is investigated. 5. Results and discussions In this section the results for the scheme proposed in the previous sections are presented in two parts. Section 1 deals with the analysis of the average buffer delay using the Green’s function method presented in Section 4 and compares the results against the methods discussed in Refs. [16,17]. The second part discusses the capacity improvement that can be achieved using the probability transmission and threshold buffer management scheme against the simultaneous transmission scheme in Ref. [1]. 5.1. Average packet delay analysis The arguments presented in Section 4 imply that the analytical methods suggested in Refs. [16,17] are

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Fig. 13. Average packet delay versus average arrival rate.

inadequate to accurately determine the average packet delay of the threshold based buffer management scheme presented in Section 4. In this section we present results comparing the approach suggested in this paper with the approaches in Refs. [16,17] and simulation results. The authors in Refs. [16,17] present analysis for the case when the transmission servers are always available (i.e. pT1 ¼ pT2 ¼ 1), and T1 ¼ 0 and T1 and T2 can take any values. Using the same assumptions for the model in this paper, Fig. 13 shows the average packet delay for all approaches and the simulations for the special case of pT1 ¼ pT2 ¼ 1; T1 ¼ 0; T2 ¼ 10; RT1 ¼ 1; RT2 ¼ 2: Fig. 13 shows that the method proposed in this paper is better able to predict the average packet delay for the threshold based buffer management system in Fig. 9. The method in Ref. [16] performs worse, as it is unable to correctly analyse the bulk arrival case. All the curves in Fig. 13 tend to infinity at l ¼ 2 as at that point the arrival rate exceeds the transmission rate. 5.2. Results and discussions for capacity comparison A Monte-Carlo simulation is used to compare the performance of the simultaneous transmission system discussed in Section 2 and the probability transmission scheme using the threshold based buffer management scheme. The simulations are used to compare how the average total transmission power varies as a function of the total transmission rate delivered by the base station to all the users in the system, as well as the average packed delay encountered by users in both schemes. We assume 20 users in the cell ðN ¼ 20Þ; uniformly, randomly distributed across the cell. The path gain from the base station to each user is calculated using the inverse fourth power law [7]. All users demand an equal bit rate that comes from the allowable set of OVSF codes, hence ri [ {16, 32, 64, 128, 256, 512, 1024, 2048, 4096} kb/s or Ri [ {1, 2, 4, 8, 16, 32, 64, 128, 256} packets/frame. Inter-cell interference is assumed constant for all users as 0.1 W and an orthogonality factor of 0.6 with

Fig. 14. Average transmitted power per frame versus average transmission rate.

a target Eb =N0 of 4 dB. The value of normal channel noise used is N0 ¼ 10212 W/Hz, and the bandwidth W ¼ 4:096 MHz. Fig. 14 shows the total base station transmission power versus the total transmission bit rate for the simultaneous transmission scheme and the probability scheme for different values of Pr : The figure shows that using the probability transmission scheme, it is possible to improve the maximum system capacity. The improvement in system capacity is dependent on Pr and there exist an optimal value of Pr for which the improvement in system capacity is maximised and calculated as Pr ¼ 0:1 from Eq. (19). As described in Section 2, this improvement in maximum capacity is obtained by reducing the average number of users transmitting simultaneously. As Nav ¼ Pr N; this has the effect of reducing the self-interference in the cell. Fig. 15 shows the improvement in total transmission rate versus transmission power for varying values of Pr : Fig. 15 shows that for the system parameters chosen, it is possible to obtain capacity improvement of over 40% for a transmission probability of 0.1. However, this improvement in system capacity in Fig. 15 comes with a penalty of an increase in

Fig. 15. Percentage rate improvement versus average transmitted power.

O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

Fig. 16. Average packet delay versus average transmitted power.

average packet delay. This increase in average packet delay is shown in Fig. 16. The best delay performance is obtained in the simultaneous transmission scheme, and the worst performance is obtained when transmissions occur with the optimum probability Pr ¼ 0:1: However, so far as the average packet delay is kept within the allowable delay constraints the system performance is tolerable. The 3G standards [19] suggest that the maximum tolerable end-to-end delay for conversation class services (voice and video telephony) is less than 100 ms, streaming class services (web broadcasts and video streaming on demand) is less than 150 ms and interactive class services (web browsing) is less than 300 ms. With the length of a W-CDMA frame as 10 ms, it implies that for a transmission probability Pr ¼ 0:1; the scheme would be best used for the streaming and interactive classes of service, where about 53 and 26%, respectively, of the allowable average packet delay must be spent in the buffer. For conversational class services, this percentage would be as high as 80%. Another option would be to use a greater transmission of Pr ¼ 0:4; which corresponds to a reduced capacity improvement of about 20-25%, but less than half the increase in average packet delay obtained when transmitting with a probability Pr ¼ 0:1: It has been assumed in this paper that all users have the same transmission probability Pr . However, in practice, different users in the cell will require different classes of service and can thus be assigned different probabilities to optimise the capacity of the wireless link while still meeting all QoS targets. 6. Conclusions In this paper we present a capacity improvement scheme for W-CDMA systems based on a probability transmission scheme in conjunction with a threshold based buffer management scheme. We show that this scheme is able to deliver significant capacity improvement over the simultaneous transmission scheme.

1307

This paper shows that for the simultaneous transmission scheme in Ref. [1], there is a constraint on the maximum total bit-rate that can be accommodated on the downlink of W-CDMA systems, even in the case of infinite power. This limit in total system capacity is due to the self-interference generated by the simultaneous transmissions to users. We show that this system capacity is a function of the number of users N, simultaneously transmitting in the cell. Eq. (15) shows the limiting value of the total transmission rate, and suggests that if the average number of users that simultaneously transmit can be kept low ðNav , NÞ; it would be possible to improve the system capacity. We achieve this reduction in the average number of transmitting users by allowing each user access to the channel with a probability Pr . Section 2.3 shows that to maximise the system capacity, an optimal transmission probability exists and can be calculated using Eq. (19). For the system under consideration in this paper, this optimum value of Pr is about 0.1. Since each user is to be allowed access to the channel every 1/Pr frames on average, to avoid excess packet delay, a method is proposed for matching any arrival rate to a set of discrete transmission rates such that the allocated rates and probabilities are able to cope with the accumulation of packets when there are no transmissions. Because of the OVSF code tree structure, transmission rates can only be allocated in discrete steps, increasing in powers of two. This represents wastage of resources, as the system might allocate a transmission rate that exceeds the arrival rate requirement. To ensure that each transmission can occur only when enough packets are queued and avoid wastage of resources, a threshold based buffer management scheme is implemented and is shown in Fig. 9. Simulations show that using the buffer management scheme ensures that each user’s transmissions occur with a probability Pr along with the matching of the arrival rate to the allocation of transmission rates as suggested in Section 3. Because using the probability transmission and buffer management schemes requires packets to be queued in the buffers until they are allocated resources, this entails an increase in the average packet delay. An analytical expression for the average packet delay in the buffer management scheme is given, as existing methods do not provide accurate results. Simulation results show that using the resource allocation scheme proposed, it is possible to achieve 30– 40% improvement in system capacity when the user transmits at optimum probability. However this leads to an average packet delay of about 80 ms in the base station buffers. This is a significant portion of the allowable end-to-end delay of conversation class users (voice and video telephony), but could be suitable for other service classes such as streaming (web streaming) and interactive (web browsing) with less restrictive delay constraints. If the delay constraints are still too restrictive, over 20% capacity improvement can be

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O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

achieved by transmitting at Pr ¼ 0.4, with a penalty of less than 40 ms.

Acknowledgements This work is supported by Motorola UK. The authors would like to acknowledge the discussions with Drs Luis Lopes, Eric Villier and Walter Featherstone at Motorola UK. The authors would also like to thank S. Kahn and M.A. Imran of Imperial College, London for their valuable comments and help with this paper.

Appendix A This appendix gives the derivation of the analytical expression for the average packet delay of the threshold based buffer management scheme depicted in Fig. 9. The Markov chain representation of Fig. 9 is shown in Fig. A1. The Markov chain is shown in three layers to clearly represent the three individual buffers, i.e. 1. Process 0: 0 to T1 2 1; with no transmission rate allocated, 2. Process 1: T1 to T2 2 1; with server rate RT1 and probability pT1 ; and 3. Process 2: T2 to 1, with server rate RT2 and probability pT2 : The transition probabilities in each process i are labelled as ai;x ; where x is the difference between the two states. Since the server rates, server probabilities and packet arrival probabilities are known, the transition probabilities can be calculated. For instance, assume that the probability of l packets arriving in a frame is al and packets can be

transmitted at a rate R with probability p: The transition probability from a state n 2 x to a state n is the probability that no packets are transmitted and exactly x packets arrive plus the probability that R packets are transmitted but x þ R packets arrive during that frame, i.e. Probðn; n 2 xÞ ¼ ax ð1 2 pÞ þ axþR p: The ergodic Green’s function for all the processes is determined below as in Appendix A.1. A.1. Ergodic Green’s functions A.1.1. z-Transform of ergodic Green’s function for process 0 Writing the equation governing the transition of Process 0, similar to Eq. (28) gn;0 ðk þ 1Þ ¼

1 X

gn2x;0 ðkÞax :

ðA1Þ

x¼0

The term gx;y represents the ergodic process of state x in process y: We find the ergodic Green’s function of Eq. (A1) " #  m m 1   X X X   gn;0 ðk þ 1Þ ¼ gn2x;0 ðkÞax  ; ðA2Þ  

k¼0

k¼0

m!1

gn;0 2 gn;0 ð0Þ ¼

1 X x¼0

"

m X k¼0

x¼0

m!1

#  1  X  ax ¼ gn2x;0 ðkÞ gn2x;0 ax : m!1 x¼0

ðA3Þ Then, taking the z-transform of both sides gives the following " # 1 1 1 1 X X X X n n gn;0 z 2 gn;0 ð0Þz ¼ gn2x;0 ax zn ; ðA4Þ n¼21

n¼21

Fig. A1. Markov chain representation of Fig. 9.

n¼21

x¼0

O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

G0 ðzÞ 2 1 ¼

1 X

"

#

1 X

gn2x;0 zn2x ax zx ¼ G0 ðzÞ

n¼21

x¼0

1 X

G1 ðzÞ21¼

ax z x

x¼2RT1

x¼0

ðA5Þ

¼ G0 ðzÞAðzÞ;

"

1 X

þ

1 X

"

1309

#

1 X

gn2x;1 z

n2x

axþRT1 zx pT1

n¼21 1 X

# gn2x;1 z

n2x

ax zx ð12pT1 Þ;

ðA11Þ

x¼0 n¼21

1 : G0 ðzÞ ¼ 1 2 AðzÞ

G1 ðzÞ21¼G1 ðzÞz2RT1

1 X

axþRT1 zxþRT1 pT1

x¼2RT1

Appendix A.1.2. z-Transform of ergodic Green’s function for process 1 Similarly, we begin with the equation governing the transition of Process 1 1 X

gn;1 ðk þ 1Þ ¼

þG1 ðzÞ

1 X

ax zx ð12pT1 Þ;

ðA12Þ

x¼0

G1 ðzÞ¼

ðz

RT1

ðzRT1 =AðzÞÞ : =AðzÞÞ2ð12pT1 ÞzRT1 2pT1

ðA13Þ

gn2x;1 ðkÞaxþRT1 pT1

x¼2R1

þ

1 X

gn2x;1 ðkÞax ð1 2 pT1 Þ:

ðA6Þ

x¼0

We can then find the ergodic Green’s function of both sides of Eq. (A6). " #  m m 1  X X X   gn;1 ðk þ 1Þ ¼ gn2x;1 ðkÞaxþRT1 pT1   m!1 k¼0 k¼0 x¼2RT1 m!1 " # m 1  X X  þ gn2x;1 ðkÞax ð1 2 pT1 Þ  ; ðA7Þ  k¼0

x¼0

A.1.3. z-Transform of ergodic Green’s function for process 2 Similarly to the derivation of Eq. (A13), the z-transform of the Ergodic Green’s function of process 2 is obtained as G2 ðzÞ ¼

ðzRT2 =AðzÞÞ : ðzRT2 =AðzÞÞ 2 ð1 2 pT2 ÞzRT2 2 pT2

Appendix A.2. Compensating functions for process 0

m!1

"

1 X

gn;1 2gn;1 ð0Þ¼

x¼2RT1

The transition equation governing the infinite chain of Process 0 is given as

#  m  X  gn2x;1 ðkÞm!1 axþRT1 pT1 

pðn;0Þ ¼

k¼0

#  1 m  X X  þ gn2x;1 ðkÞm!1 ax ð12pT1 Þ;  "

x¼0

1 X

gn;1 2gn;1 ð0Þ¼

gn2x;1 axþRT1 pT1 þ

1 X

1 X

pðn2x;0Þ ax ;

ðA15Þ

x¼0

ðA8Þ

k¼0

x¼2RT1

gn2x;1 ax ð12pT1 Þ:

where pðx;yÞ represents the probability of the queue being in state x in process y: Then, the transition equations governing the infinite chains for Processes 1 and 2 are given as below pðn;1Þ ¼

1 X

pðn2x;1Þ axþRT1 pT1 þ

x¼2RT1

x¼0

1 X

pðn2x;1Þ ax ð1 2 pT1 Þ;

x¼0

ðA16Þ

ðA9Þ Taking the z-transform of both sides of Eq. (A9), gives the ergodic Green’s function 1 X

¼

2 4

n¼21 1 X

gn2x;1 axþRT1 pT1 5zn

1 X

n¼21 x¼0

pðn2x;2Þ axþRT2 pT2 þ

x¼2RT2

pð0;0Þ ¼ pð0;0Þ a0 ;

x¼2RT1

"

1 X

1 X

pðn2x;2Þ ax ð1 2 pT2 Þ;

x¼0

The forward Kolmogorov equations for the Markov chain shown in Fig. 17 are written for each state n as follows:

3

1 X

pðn;2Þ ¼

ðA17Þ gn;1 ð0Þzn

n¼21

1 X

þ

1 X

gn;1 zn 2

n¼21

ðA14Þ

# gn2x;1 ax ð12pT1 Þ zn ;

ðA10Þ

pðn;0Þ ¼

n X x¼0

n ¼ 0;

pðn2x;0Þ ax ;

1 # n # T1 2 RT1 ;

ðA18Þ ðA19Þ

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O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

n X

pðn;0Þ ¼

pðn2x;0Þ ax þ

A.3. Compensating functions for process 1

n2T X1

pðn2x;1Þ axþRT1 pT1

x¼2RT1

x¼0

ðA20Þ cðn;1Þ ¼ 2

T1 2 RT1 , n , T1 ; n X

pðn;1Þ ¼

n2T X1

pðn2x;0Þ ax þ

n2T X1

pðn2x;1Þ axþRT1 pT1 ;

ðA27Þ

x¼RT1

x¼n2T1 þ1

þ

n2T X1

T1 2 RT1 # n , T1 ; pðn2x;1Þ axþRT1 pT1

x¼2RT1

ðA21Þ pðn2x;1Þ ax ð1 2 pT1 Þ;

n X

cðn;1Þ ¼

x¼0

pðn2x;0Þ ax

T1 , n , T2 2 RT2 ;

ðA28Þ

x¼n2T1

T1 # n , T2 2 RT2 ; n X

pðn;1Þ ¼

n2T X1

pðn2x;0Þ ax þ

x¼n2T1 þ1

þ

n2T X1

pðn2x;1Þ axþRT1 pT1

x¼2RT1 n2T X2

pðn2x;2Þ axþRT2 pT2

pðn2x;2Þ axþRT2 pT2 ;

x¼2RT2

ðA29Þ

T2 2 RT2 # n , T2 ;

x¼2RT2

T2 2 RT2 # n , T2 ; n X

ðA22Þ

n2T X1

pðn2x;0Þ ax þ

x¼n2T1 þ1

þ

n2T X2

pðn2x;0Þ ax þ

x¼n2T1

pðn2x;1Þ ax ð1 2 pT1 Þ þ

x¼0

pðn;2Þ ¼

n X

cðn;1Þ ¼

n2T X1

pðn2x;1Þ axþRT1 pT1 n2T X2

pðn2x;1Þ ax ð1 2 pT1 Þ þ

x¼2RT2

 pðn2x;2Þ axþRT2 pT2 þ

n2T X2

pðn2x;1Þ axþRT1 pT1

x¼2RT1

x¼2RT1

x¼0

n2T X1

cðn;1Þ ¼ 2

2

n2T X1

pðn2x;1Þ ax ð1 2 pT1 Þ;T2 . n:

ðA30Þ

x¼0

ðA23Þ A.4. Compensating functions for process 2

pðn2x;2Þ ax ð1 2 pT2 Þ;

x¼0

n . T2 :

cðn;2Þ ¼ 2

For each state x in process y; the compensating function cx;y can be obtained by subtracting the transition equations of chain B from the appropriate chain A equation. Hence, for each process, the compensating functions are determined as shown below. Process 0 n # 0 # T 2 RT1 ;

cðn;0Þ ¼ 0;

cðn;0Þ ¼

n2T X1

ðA24Þ

pðn2x;1Þ axþRT1 pT1 ;

pðn2x;2Þ axþRT2 pT2

x¼2RT2

ðA31Þ

T2 2 RT2 # n , T2 ;

cðn;2Þ ¼

n X x¼n2T1

þ

x¼RRT1

n2T X2

n2T X1

pðn2x;0Þ ax þ

n2T X1

pðn2x;1Þ axþRT1 pT1

x¼2RT1

ðA32Þ pðn2x;1Þ ax ð1 2 pT1 Þ

x¼0

ðA25Þ

T2 . n:

T1 2 RT1 # n , T1 ; A.5. Determining the simultaneous equations cðn;0Þ ¼ 2

n X x¼0

pðn2x;0Þ ax ;

n $ T1 :

ðA26Þ

Using Eqs. (A24) – (A32), the z-transforms of the compensating functions for each of the processes can be

O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

obtained as follows: "

T1 X

C0 ðzÞ ¼ P0

n X

p0;ðnÞ 2

n¼T1 2RT1

n¼T1

n X

#

"

T2 X

þ

p0;ðn2xÞ ax zn ;

ðA33Þ

2

m2T X1 x¼0

þ

C1 ðzÞ ¼PT1

42

n¼T1 2RT1

2

þ PT1

2

n¼T2

2

n2T X1

n X

p1;ðnÞ 2

"

1 X

m2T X1

p1;ðm2xÞ axþRT1 pT1

x¼2RT1

#

p1;ðm2xÞ ax ð1 2 pT1 Þ ;

T 1 # n , T2 :

ðA37Þ

x¼0

p1;ðn2xÞ axþRT1 pT1

x¼n2T1

#

p1;ðn2xÞ ax ð1 2 pT1 Þ zn

x¼0

#

2

m2T X1

p1;ðm2xÞ axþRT1 pT1

x¼m2T1

x¼2RT1

n¼T2 2RT2 n2T X1

2

p1;ðn2xÞ axþRT1 pT1 5zn

"

T2 X

þ PT1

3

n2T X1

m X

p1;ðm2xÞ ax ð1 2 pT1 Þ gðn2m;1Þ

m¼T2

2

p1;ðm2xÞ axþRT1 pT1 5gðn2m;1Þ

p1;ðmÞ 2

"

1 X

3

x¼2RT1

m¼T2 2RT2

x¼0

T1 X

n2T X1

42

m¼T1 2RT1

p0;ðn2xÞ ax zn

x¼0

"

1 X

þ P0

p1;ðnÞ ¼

#

2

T1 X

1311

n2T X1

p1;ðn2xÞ axþRT1 pT1

x¼2RT1

Eq. (A36) gives T1 equations with T1 unknowns for the values of p0;ðnÞ : These equations can be solved simultaneously to determine the values of p0;ðnÞ : This is similarly represented in Eq. (A37) where T2 2 T1 equations are present with T2 2 T1 unknowns. The values of gðn;iÞ can be obtained iteratively from Eqs. (A3) and (A9), so that ðkþ1Þ ¼ gðn;iÞ ð0Þ þ gðn;iÞ

#

1 X

gðkÞ ðn2x;iÞ ax ;

i ¼ 0;

ðA38Þ

x¼0

p1;ðn2xÞ ax ð1 2 pT1 Þ zn ;

ðA34Þ

x¼0 ðkþ1Þ gðn;iÞ ¼ gðn;iÞ ð0Þ þ

1 X

gðkÞ ðn2x;1Þ axþRTi pTi

x¼2RTi

2

T2 X

C2 ðzÞ ¼ PT2

42

n¼T2 2RT2

2

p2;ðn2xÞ axþRT2

n2T X2

p2;ðnÞ 2

n¼T2 n2T X2

3

þ

pT2 5zn

p2;ðn2xÞ axþRT2 pT2

x¼2RT2

p2;ðn2xÞ ax ð1 2 pT2 Þ zn ;

ðA35Þ

x¼0

where piðnÞ is the conditional probability of the system having process i active with n packets in the system. P0 ; PT1 and PT2 are the probabilities of the queue being in process 0, process 1 or process 2, respectively. We recall from Eq. (37) that the probabilities can be written in convolution terms as

p0;ðnÞ ¼

" p0;ðnÞ 2

m¼T1 2RT1

þ

1 X m¼T1

"

m X x¼0

gðkÞ ðn2x;1Þ ax ð1 2 pTi Þ;

i ¼ 1;

ðA39Þ

x¼0

#

T1 X

1 X

x¼2RT2

"

1 X

þ PT2

n2T X2

m X

# p0;ðm2xÞ ax gðn2m;0Þ

x¼0

#

p0;ðm2xÞ ax gðn2m;0Þ ; 0 # n , T1 ;

ðA36Þ

where k is the iterative index and gðn;iÞ ð0Þ ¼ 1 if n ¼ 0; otherwise gðn;iÞ ð0Þ ¼ 0: In practice, it is necessary to truncate n and k to finite values. The conditional probabilities for p2;ðnÞ can either be obtained in a similar manner to p0;ðnÞ and p1;ðnÞ or by approximating the probabilities p2;ðnÞ as the probabilities from an infinite length buffer served with a server of rate RT2 at a probability pT2 as shown in Appendix B. A.6. Combining the three buffer analysis The values of P0 ; PT1 and PT2 ; which are defined as the probability of being in either process 0, 1 or 2, respectively, can be determined by aggregating these individual processes into a three state Markov chain as shown in Fig. A2. Pxy is transition probability from process x to process y: As the conditional probabilities for all processes are now known, Pxy can be calculated using the arrival process. Once obtained, P0 ; PT1 and PT2 are determined using the standard queuing theory analysis. The probability generating function of the whole system can be

1312

O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

Fig. A2. Markov chain aggregate of process 0, 1 and 2.

stated as PðzÞ ¼ P0

Eq. (B2) becomes: TX 1 21 n¼0

! p0;ðnÞ z

n

0 þ PT1 @

TX 2 21

1 p1;ðnÞ z

nðzÞ ¼ E½zn  ¼ E½zaþðn2sÞUðn2sÞ :

nA

Since a;n and s are independent variables we have

n¼T1

0

1

RY B C T2 21 Cðz 2 1Þ B C ðz 2 zR ÞC: þ PT2 B RT2 @z A 2 ð1 2 pT2 ÞzRT2 2 pT2 R¼1 AðzÞ ðA40Þ

Assuming that the arrival process is Poisson, PðzÞ can thus be expressed as follows: 0 1 ! TX TX 1 21 2 21 n n PðzÞ ¼ P0 p0;ðnÞ z þ PT1 @ p1;ðnÞ z A n¼0

n¼T1

{ðRT2 2 lÞ 2 RT2 ð1 2 pT2 Þ}ðz 2 1Þ zRT2 2 ð1 2 pT2 ÞzRT2 2 pT2 AðzÞ  ! RY T2 21 z 2 zR :  1 2 zR R¼1

ðB3Þ

nðzÞ ¼ E½za £E½zðn2sÞUðn2sÞ  ¼ AðzÞE½zðn2sÞUðn2sÞ :

Considering the second term of the right hand side of Eq. (B4) and taking the expectation, first of n and then of s; it may be rewritten as E½zðn2sÞUðn2sÞ  ¼ Es ½En ½zðn2sÞUðn2sÞ  9 28 3 1
ðn2sÞUðn2sÞ

 ¼ Es 4Pr½n # sþ

ðB5Þ 3

1 X

Pr½n ¼ jz

j2s 5

j¼sþ1

2

þ PT2

ðB4Þ

1 X

¼ Es 4 P0 þ

3

j2s 5

pj z

;

ðB6Þ

j¼sþ1

ðA41Þ

where P0 is equivalent to PðkÞ in the limit as k !1: PðkÞ is defined as the probability that there are less that sðkÞ packets in the queue at the start of the kth time frame and pj is the probability of j packets in the queue. Then, we can say PðkÞ ¼ Pr½nðkÞ # sðkÞ ¼

Appendix B

s X

pr :

ðB7Þ

r¼0

In this appendix we derive pgf of an infinite length discrete queue with a server available with a finite probability less than 1. The dynamic state of the system queue is expressed as nðk þ 1Þ ¼ aðkÞ þ ðnðkÞ 2 sðkÞÞUðnðkÞ 2 sðkÞÞ;

ðB1Þ

where at the kth time frame, nðkÞ is the queue length; aðkÞ; the number of packets that arrive to the queue; sðkÞ is the server rate. UðtÞ is the Heaviside function, defined as UðtÞ ¼ 1 for t $ 0 and 0 otherwise. Using Eq. (B1) the pgf is expressed as follows: nðk þ1ÞðzÞ ¼ E½znðkþ1Þ  ¼ E½zaðkÞþðnðkÞ2sðkÞÞUðnðkÞ2sðkÞÞ :

Eq. (B.6) can be rewritten as follows: 8 93 2 3 2 1 1 s
ðB2Þ

By considering the limit where k !1; the system is in steady state and the dependence on k may be suppressed. Thus dropping the index, the limiting distribution of

ðB8Þ Simplifying Eq. (B8) gives 2 3 s X 2s j2s Es ½z nðzÞ þ Es 4P0 2 pj z 5 j¼0

2 3   s s X X 1 ¼S nðzÞ þ Es 4 pj 2 pj zj2s 5: z j¼0 j¼0

ðB9Þ

Given that the maximum server rate is RT2 packets/frame, let sR be the probability that the server rate is R: The second

O.O. Oyefuga et al. / Computer Communications 27 (2004) 1295–1313

term of right hand side of Eq. (B9) can be written as 2 3 RT2 RT2 s s R R X X X X X X Es 4 pj 2 pj zj2s 5 ¼ s R pj 2 sR pj zj2R j¼0

R¼0

j¼0

R¼0

j¼0

Assuming a Poisson arrival process and using the property that Qð1Þ ¼ 1 and applying L’Hopital’s rule [14], C is found as

j¼0

ðB10Þ and simplifying to give z2RT2

RT2 21 X

pj

RT2 X

RX T2 21

sR {zRT2 2 zjþRT2 2R }:

pj

RT2 X R¼j

j¼0

zRT2

:

ðB17Þ

ð1 2 zR Þ

ðB11Þ

sR {zRT2 2 zjþRT2 2R }  ! 1 1 2S AðzÞ z

:

ðB12Þ

The above equation involves the calculation of RT2 probabilities. However one can avoid this calculation by examining the bounds of nðzÞ: It can be shown that under the condition of stochastic equilibrium, i.e. where the channel utilization, r ¼ ðl=SÞ , 1; the denominator of nðzÞ has exactly RT2 2 1 zeros z1 ; z2 ; …;zRT221 in the unit circle lzl , 1 of the complex plane. Under the assumption of a Poisson arrival process, Eq. (B12) becomes: RX T2 21 j¼0

pj

RT2 X

sr {zRT2 2 zjþRT2 2R }

R¼j

zRT2 e2lðz21Þ 2 zRT2 S

 ! : 1 z

ðB13Þ

There are RT2 unknowns, pj ; 0 # i # RT2 2 1: To determine these unknown probabilities we appeal to the bounds of nðzÞ inside the unit circle based on Rouche’s theorem in Ref. [15]. Using this, and because nðzÞ is analytic, when the denominator is zero the numerator must also be zero. Thus, all the roots (zeros) of the denominator must also make the numerator vanish. Since the numerator of Eq. (B13) is aQpolynomial of T2 21 degree RT2 ; then it can be written as Cðz 2 1Þ RR¼1 ðz 2 zR Þ; where C is a constant. Using this nðzÞ can be written as nðzÞ ¼ zRT2

RY T2 21 Cðz 2 1Þ  ! ðz 2 zR Þ: 1 1 R¼1 2S AðzÞ z

ðB14Þ

Given that in Process 2 in Appendix A has a server with rate RT2 with a probability pT2 the z-transform of the server process can be written as SðzÞ ¼ pT2 zRT2 þ ð1 2 pT2 Þ:

ðB15Þ

Hence, the pgf of the probabilities of process 2 in Appendix A can be expressed as nðzÞ ¼

1

R¼1

Combining Eqs. (B.4), (B.9) and (B.11) the pgf is given by

nðzÞ ¼

C ¼ {ðRT2 2 lÞ 2 RT2 ð1 2 pT2 Þ} R 21 t2 Y

R¼j

j¼0

nðzÞ ¼

1313

Cðz 2 1Þ

RY T2 21

z 2 ð1 2 pT2 ÞzRT2 2 pT2 AðzÞ

R¼1

RT2

ðz 2 zR Þ:

ðB16Þ

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