design and performance modeling of resource allocation strategies for

Journal of the Chinese Institute of Engineers, Vol. 31, No. 3, pp. 385-401 (2008)

385

DESIGN AND PERFORMANCE MODELING OF RESOURCE ALLOCATION STRATEGIES FOR GPRS

Yi-Chou Tsai*, Huei-Wen Ferng, and Jeng-Ji Huang

ABSTRACT Incorporating promising techniques, including buffering, priority, de-allocation, preemption, and threshold control on the buffer into the resource allocation design for the general packet radio service (GPRS) network, four resource allocation strategies are proposed in this paper, i.e., strategies RAS 1 , RAS 2, RAS th1, and RAS th2 . For the four strategies, analytic models are built to obtain performance measures expressed in general forms using the Markov chain approach. In addition, simulation experiments are arranged and used to validate the analytic results. Based on the numerical results, we show that i) a good match between analytic and simulation results strongly supports the theoretical analysis given in this paper; ii) these four strategies, designed in a comprehensive manner, can outperform many strategies previously proposed in the literature; iii) better voice call performance and superior quality of service (QoS) differentiation between new and handoff voice calls can be achieved by using these strategies. Key Words: general packet radio service (GPRS), resource allocation, de-allocation, preemption.

I. INTRODUCTION Contemporary and future mobile networks are designed toward supporting multimedia services rather than a single voice service as wireless technologies, e.g., the third generation (3G) standard (Dahlman et al., 1998), (ETSI, 2002), (ETSI, 2003a), (ETSI, 2003b), (Holma and Toskala, 2002) and GPRS (ETSI, 1999) etc., advance and the population of subscribers increases. Overlaid over the second generation (2G) systems, e.g., GSM (Eberspacher et al., 2001) and IS-136 (Sollenberger et al., 1999), GPRS (ETSI, 1999), which is an intermediate system between 3G and 2G systems, is capable of carrying multimedia services and has drawn much research attention. Different from GSM and IS-136, GPRS is designed based on packet-switching instead of circuit-switching to *Corresponding author. (Tel: 886-2-25337698; Email: yichou. [email protected]) Y. C. Tsai and H. W. Ferng are with the Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan, R.O.C. J. J. Huang is with the Institute of Applied Electronics Technology, National Taiwan Normal University, Taipei 106, Taiwan, R.O.C.

afford mobile users multimedia services. Since the common air interface is shared by voice and data users/services, how to properly manage radio resources to satisfy different QoS requirements requested by multimedia services has become a hot research issue in the literature. In this paper, we touch the issue through exploring resource allocation strategies for GPRS using a comprehensive approach. To understand the GPRS system as shown in Fig. 1, let us briefly explain the following two parts, i.e., wired and wireless parts. For more details, readers can refer to (ETSI, 1999), (Lin and Lin, 2001) and references therein. In the wired part, two extra nodes, i.e., one serving GPRS support node (SGSN) and one serving gateway GPRS support node (GGSN), as compared to the GSM system, are endowed with GPRS to handle routing of data packets to a mobile station (MS) and to an external data network, respectively. As for the wireless part, GPRS shares a common air interface with GSM. For GPRS, the channel dedicated to GPRS data traffic is called packet data channel (PDCH). Let us now examine the procedure of channel access and allocation for the uplink data packet transfer. An MS going to initiate data packet transfer first sends a request on the packet random

386

Journal of the Chinese Institute of Engineers, Vol. 31, No. 3 (2008)

PSTN

MSC/VLR

HLR

SGSN

GGSN

BSS BTS

BSC

PDNs : Signalling interface : Signalling and data transfer interface

Other PLMN GGSN

Fig. 1 The architecture of GPRS

access channel (PRACH). Then, the packet assignment is responded to on the packet access grant channel (PAGCH) to inform the MS of channel reservation. Of course, the MS may further send another request based on the data traffic load to invoke reallocation on the packet associated control channel (PACCH). The above procedure illustrates the flexible and dynamic nature of channel/resource allocation for GPRS. In fact, more than one physical channel may be allocated to a GPRS data user or more than one data user can share one physical channel based on the capacity-on-demand principle (ETSI, 1999). Generally speaking, such allocation schemes/ strategies are frequently called dynamic resource allocation (DRA) or dynamic channel allocation (DCA) schemes/strategies. In the literature, various techniques have been proposed and studied to improve performance of a cellular system, including queueing/buffering (Hong and Rappaport, 1986), (Lin et al., 1994), cut-off priority or reservation (Hong and Rappaport, 1986), (Oh and Tcha,1992), and preemption (Garay and Gopal, 1992), (Wang et al., 2003), etc. For GPRS, de-allocation of PDCHs and service priority are also recommended by (ETSI, 1999). Utilizing any one or a mixture of the aforementioned techniques, some papers have discussed the issue of design and performance of DRA/DCA schemes, e.g., (Chen et al., 2003), (Ferng and Tsai, 2005), (Lin and Lin, 2001), (Lin, 2003), (Liu et al., 2002), (Zhang and Soong, 2004), (Zheng and Regentova, 2004) etc. Chen et al. and Zheng and Regentova proposed de-allocation mechanisms which may de-allocate some channels occupied by a data user when no available channel is found upon a voice user/call arrival. In (Chen et al., 2003), simple de-allocation analytic models are given, while Zheng and Regentova further studied two new de-allocation schemes based on (Chen et al., 2003). Ferng and Tsai investigated the effects of various

techniques, including priority, buffering, threshold control on the voice buffer, and reservation, and proposed four DCA schemes utilizing these techniques. In (Lin and Lin, 2001), Lin and Lin studied the performance of four proposed DRA schemes based on buffering for new/handoff voice calls only. Further considering buffering for data packets and priorities for the voice and data buffers, (Lin and Lin, 2001) is extended by Lin to form (Lin, 2003). By Liu et al. adopted queueing and reservation, the mean delay time is analyzed in (Liu et al., 2002). In (Zhang and Soong, 2004), Zhang and Soong studied the re-allocation scheme for data users using an analytic approach. Regarding the above mentioned papers, (Chen et al., 2003), (Lin and Lin, 2001), (Zhang and Soong, 2004), and (Zheng and Regentova, 2004) focused on mechanisms composed of a specific technique, e.g., de-allocation etc. Although (Lin, 2003) utilized buffering and priority and (Liu et al., 2002) applied buffering and reservation to design resource/channel allocation schemes for GPRS, their designs do not incorporate all promising techniques into consideration. Furthermore, the issue of QoS is not well managed by (Lin, 2003) and (Liu et al., 2002). In contrast to (Lin, 2003) and (Liu et al., 2002), our previous work (Ferng and Tsai, 2005) devised DCA schemes by taking buffering, priority, reservation, and threshold control into consideration to differentiate/guarantee QoS of different calls. However, some promising techniques, e.g., de-allocation and preemption, are not taken into consideration by (Ferng and Tsai, 2005) to further improve performance of voice calls. Hence, we discuss this issue in this paper so that (Ferng and Tsai, 2005) is extended. Note that the reservation technique is not employed in our current design due to lack of flexibility as illustrated in (Ferng and Tsai, 2005). Also, we abandon the technique of re-allocation (Zhang and Soong, 2004) because of its marginal contribution if buffering is considered. Thus, we propose several dynamic resource allocation strategies in this paper based on buffering (with or without threshold control), priority, de-allocation, and preemption. Moreover, general and analytic results are provided for these proposed strategies to facilitate system design through the Markovian analytic approach. To validate the effectiveness of these strategies, a selfsimilar traffic model (Ferng and Chang, 2001), (Leland et al., 1994), (Paxson and Floyd, 1995) for data traffic is employed as well to have a comparison with results based on the Poissonian data traffic model. The proposed four strategies can exhibit better QoS for voice calls with a little degradation for data packets and perform better than strategies previously proposed in (Ferng and Tsai, 2005), (Lin and Lin, 2001), (Lin, 2003).

Y. C. Tsai et al.: Design and Performance Modeling of Resource Allocation Strategies for GPRS

The rest of the paper is organized as follows. We discuss the design of dynamic resource allocation strategies in Section II and the corresponding performance analysis in Section III and the Appendix. In Section VI, the performance of different DRA strategies is examined through extensive numerical examples. Finally, Section V concludes the paper. II. DESIGN OF DYNAMIC RESOURCE ALLOCATION STRATEGIES Based on buffering with/without threshold control on the voice buffer, priority, de-allocation, and preemption techniques, we design four dynamic resource allocation strategies under some common assumptions: i) the number of radio channels shared by both voice and data users is fixed at C; ii) one channel is allocated upon the request of a voice user, while at most n channels may be allocated to the request of a data user. Note that the above two assumptions were adopted in (Ferng and Tsai, 2005) as well. For the four proposed strategies, buffering, priority, and de-allocation are utilized to form the substrata. Hence, there are two different buffers for each strategy: a voice buffer with size B v to accommodate new/originating and handoff GSM voice calls and a data buffer with size B d to accommodate GPRS data packets. And, two types of priority are set for each strategy, i.e., priorities among buffers and service priorities. As far as the first type of priority is concerned, the voice buffer is allowed to possess a higher priority than the data buffer, while handoff voice calls in the voice buffer have higher precedence over new voice calls in the voice buffer in the second-type priority setting. Moreover, de-allocation works as follows: one of the channels occupied by a data user with the most occupied channels (randomly select one if multiple such data users exist) is released to a voice user if no available channel is found upon its arrival. Note that de-allocation is not allowed to perform when each data user entering the system uses one channel to guarantee the minimum data rate. For such a situation, voice calls are then temporarily queued in the voice buffer. Integration of buffering, priority, and de-allocation thus forms strategy RAS 1 . To further improve performance of voice calls, strategy RAS 2 incorporates conditional preemption into strategy RAS1. This (conditional) preemption is activated when the following conditions are true: i) no available channel is found upon arrival of voice calls; ii) each data user entering the system uses one channel only; iii) the data buffer is not full. The (conditional) preemption enables a voice call to get one channel occupied by a data user picked at random and results in that data packet being put into the data buffer for retransmission.

387

Of course, we may further consider the technique of threshold control on the voice buffer. For this technique, a threshold T (0 ≤ T ≤ B v) is specified to limit the maximum number of queued new voice calls to this value. For more details, one may refer to (Ferng and Tsai, 2005). Adding threshold control on the voice buffer to strategies RAS 1 and RAS 2 respectively produces strategies RASth1 and RAS th2. To have a direct comparison, we summarize these four strategies, i.e., RAS 1, RAS 2 , RAS th1 and RAS th2, as follows: Strategy RAS1: It is composed of buffering without threshold control on the voice buffer, priority, and de-allocation. Strategy RAS2: It is composed of buffering without threshold control on the voice buffer, priority, de-allocation, and preemption. Strategy RASth1: It is composed of buffering with threshold control on the voice buffer, priority, and de-allocation. Strategy RAS th2: It is composed of buffering with threshold control on the voice buffer, priority, de-allocation, and preemption. III. PERFORMANCE EVALUATION OF THE PROPOSED STRATEGIES Before analyzing the proposed strategies, let us first elaborate on the details of traffic and system modeling in the following. We assume that new voice calls (data packets) are generated according to a Poisson process with rate λ v ( λ d ). As for the voice call holding time, dwelling time of a voice call, and packet transmission time, they follow exponential distributions with mean 1/µv, 1/ η, and 1/µd, respectively. The above model assumptions enable one to derive the handoff rate of voice calls λh and the mean occupancy time of a voice call within a cell 1/ µ . They follow relations λ h = η (1 – Pvb ) λv /( µ v + η P ft) (Lin and Lin, 2001) and 1/µ = 1/(µ v + η ), where Pvb is the blocking probability of a new voice call and P ft is the forced termination probability of a handoff voice call. Thus, the total voice arrival rate Λ is equal to λv + λh. Note that the handoff behavior of a data user is neglected in this paper due to short transmission time of a data packet to make the resultant model simple. In fact, the assumption has been previously employed by (Ferng and Tsai, 2005) and (Lin and Lin, 2001). Finally, we denote P d and S v = (v, d 1 , ... , d n, q n , q h, qd) to represent the data packet dropping probability and a general state vector of the state space, where v, d i (1 ≤ i ≤ n), q n , q h, and q d stand for the number of voice calls in service, the number of data packets using i channels, the number of new voice calls in queue,

388


the number of handoff voice calls in queue, and the number of data packets in queue, respectively. Now, let us first detail the analysis of strategy RAS 1.

δ iδ j =

1. Analysis of strategy RAS 1 Strategy RAS1 utilizes buffering without threshold control, priorities among buffers, service priorities, and gradual de-allocation. To get system performance, we need to figure out first the steadystate probability PS v for state vector Sv in state space SRAS1 expressed as follows: S RAS1 = { S v 0 ≤ v +

n

Σ id i ≤ C , i=1

(1)

0 ≤ v ≤ C, 0 ≤ di ≤ C , i = 1, ..., n, i 0 ≤ qn ≤ B v, 0 ≤ qh ≤ B v, 0 ≤ q n + q h ≤ Bv , 0 ≤ qd ≤ B d}

n

δ pj =

Σ

i=1

δ pj =

id i = C – j

for 1 ≤ j ≤ n – 1,

0 , otherwise , n

1 , if v +

Σ

i=1

id i ≤ C – n

for j = n.

(2)

Σ di = 0 i= j+1

for 1 ≤ j ≤ n ,

n

Σ

S2 = (v + 1, d1, ... , d n, q n, qh , q d), S3 = (v, d1, ..., d n, q n – 1, qh , q d), S4 = (v, d1, ... , d n, q n + 1, q h, q d),

S6 = (v, d1, ... , d n, q n, qh + 1, q d), S7 = (v, d1, ..., d n, q n, qh, q d – 1), S8 = (v, d1, ... , d n, q n, qh, qd + 1), S9 = (v + 1, d1 – 1, ... , d n, q n, q h, qd + 1), S10 = (v – 1, d1 + 1, ..., dn, qn, qh, q d – 1), S11(s) = (v – 1, d1 + 1, ..., dn, qn + s, qh + 1 – s, qd),

S13(i) = (v, d 1 + δ 1 δi , ... , di + δ iδ i, ... , dn + δ nδ i,

S14(i) = (v – 1, d1 – δ 1δ i – 1, ... , d i –1 – δi –1 δ i –1 , di + δ iδ i, ... , d n + δ nδ i, qn, qh , qd ), S15(i) = (v + 1, d1 + δ 1δ i –1 , ..., di –1 + δi –1 δi –1 , di – δ iδ i, ... , dn – δ nδ i, qn , qh , q d).

d i = 0 for j = 0 ,

0 , otherwise . (3) The third one is δ F concerning the status of the data buffer and it is defined as follows:

δF =

S1 = (v – 1, d1, ... , d n, qn , qh , q d),

qn , q h, q d),

n

1 , if d j > 0 and i=1

For ease in balancing equations described in the following, we further define some specific types of state in shorthand as follows:

qn , q h, q d),

0 , otherwise ,

1 , if

(5)

S12(i) = (v, d 1 – δ 1δ i, ... , d i – δ iδ i, ... , dn – δ nδ i,

The second one is { δ dj |0 ≤ j ≤ n} concerning the gradual de-allocation on data packets and it can be expressed as

δ dj =

1 , if i = j , 0 , if i ≠ j .

S5 = (v, d1, ..., d n, q n, qh – 1, q d),

In fact, probabilities for states not included in S RAS1 should be set to zero so that Σ Sv ∈ SRAS P Sv = 1. Now, 1 let us further define three types of indicator function (equal to one/zero when the corresponding condition is true/false) for neater expressions. The first one is {δpj|1 ≤ j ≤ n} concerning a vector (v, d1, ..., dn) coming from the first (n + 1) elements of a source state Sv when no call/packet is queued. It satisfies the following equations: 1 , if v +

Moreover, a set of unit-length {δ j |1 ≤ j ≤ n} in which elements are pair-wise orthogonal is defined as follows:

1 , if q d = Bd , 0 , otherwise .

(4)

Using the above notations, one is able to get state transition diagrams shown in Figs. 2-3 corresponding to six classes of balance equations given by (6)(11). Now, these six classes are listed and explained as follows: Case 1: It represents the case when 0 ≤ v +

n

Σ id i < i=1

C, qn = qh = qd = 0. Considering state Sv, we note that possible events triggering state S v to change are i) an


v + 1, d1, d2, ..., dn, qn, qh, qd

v – 1, d1, d2, ..., dn, qn, qh, qd

(v + 1) µ

vµ

Λ

Λ

δ pn λd

v, d1, d2, ..., dn – 1, qn, qh, qd

δ pn λd

v, d1, d2, ..., dn, qn, qh, qd

ndn µd

v, d1, d2, ..., dn + 1, qn, qh, qd

n(dn + 1) µd

δ p1λd

δ p1λd

v, d1 – 1, d2, ..., dn, qn, qh, qd

(d1 + 1) µd

d1 µd

v, d1 + 1, d2, ..., dn, qn, qh, qd

389

change to state S v with rate Λ , ii) a data packet arrival denoting the state transition from state S 12(i) to state S v with rate δ piλd (1 ≤ i ≤ n), iii) a call completion or handoff having the state transition from state S 2 to state S v with rate (v + 1) µ , iv) channel release by a data user using i (1 ≤ i ≤ n) channels representing the state transition from state S 13 (i) to state S v with rate i(d i + 1) µ d. In Fig. 2(a), the corresponding state transition diagram is diagrammatically shown. From this figure, we can easily obtain the following balance equation by equating rates out of state S v to rates into state S v.

(a) Case 1.

(vµ + Λ +

v – 1, d1, ..., dn – 1 – 1, dn + 1, qn, qh, qd v – 1, d1 – 1, d2 + 1, d3, ..., dn, qn, qh, qd

δ dn Λ

v, d1, d2, ..., dn, qn, qh, qd + 1 λd

δ d 2Λ

v + 1, d1 – 1, d2, ..., dn, qn, qh, qd + 1

v – 1, d1, d2, ..., dn, qn, qh, qd

v µ + (q h + 1) η

v, d1, d2, ..., dn, qn, qh, qd

δpn λd

( δd0 + δd1) λv

vµ + (q n + 1)η

v, d1, d2, ..., dn, qn + 1, qh, qd

δ d 2Λ

ndn µd

v + 1, d1 + 1, d2 – 1, d3, ..., dn, qn, qh, qd

δp1λd d1 µd v, d1 – 1, d2, ..., dn, qn, qh, qd (d1 + 1) µd

δ dn Λ

v + 1, d1, ..., dn – 1 + 1, dn – 1, qn, qh, qd

(d1 + 1) µd

v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

v – 1, d1 + 1, d2, ..., dn, qn + 1, qh, qd

(b) Case 2.

v – 1, d1, ..., dn – 1 – 1, dn + 1, qn, qh, qd v , d1, d2, ..., dn, qn, qh, qd + 1

v – 1, d1 – 1, d2 + 1, d3, ..., dn, qn, qh, qd δ dn Λ λd

δ d 2Λ

(d1 + 2d2 + ... + ndn) µd ( δd 0 + δd 1) λ h

v + 1, d1 – 1, d2, ..., dn, qn, qh, qd + 1 (v + 1) µ

v , d1, d2, ..., dn, qn, qh + 1, qd v µ + (qh + 1)η (δd 0 + δd 1) λv

vµ

= ΛPS 1 + (v + 1)µ PS 2 +

v, d1, d2, ..., dn, qn, qh + 1, qd

Λ

v, d1, d2, ..., dn – 1, qn, qh, qd

(d1 + 2d2 + ... + ndn)µd ( δd 0 + δd 1) λh

(v + 1) µ

vµ

v, d1, d2, ..., dn, qn, qh, qd

v , d1, d2, ..., dn, qn + 1, qh, qd

vµ + (qn + 1)η

v – 1, d1 + 1, d2, ..., dn, qn, qh, qd – 1

δ d 2Λ v + 1, d1 + 1, d2 – 1, d3, ..., dn, qn, qh, qd

(d1 + 2d2 + ... + ndn) µd

δ dn Λ

v, d1, d2, ..., dn, qn, qh, qd – 1 λ d (d1 + 1)µd

(d1 + 1)µd

v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

n

v + 1, d1, ..., dn – 1 + 1, dn – 1, qn, qh, qd

v – 1, d1 + 1, d2, ..., dn, qn + 1, qh, qd

(c) Case 3.

Fig. 2 State transition diagrams for strategy RAS 1 (Cases 1-3)

originating/handoff voice call arrival which triggers a state transition from state Sv to state S2 with rate Λ , ii) a data packet arrival triggering a state transition from state Sv to state S13(i) with rate δpi λd (1 ≤ i ≤ n), iii) a voice call completion or handoff which makes state Sv move to state S 1 with rate vµ , iv) channel release by a data user using i (1 ≤ i ≤ n) channels, which triggers state Sv to move to state S12(i) with rate idiµd. On the contrary, the following events result in a state transition from different states to state Sv: i) an originating/handoff voice call arrival making state S 1

n

+

Σ

i=1

n

δ piλ d )PS Σ id iµ d + iΣ i=1 =1

v

n

Σ δ piλ d PS i=1

i(d i + 1)µ d PS 13(i) .

12(i)

(6) n

Case 2: It is the case when v + Σ id i = C, q n = qh = i=1 qd = 0 whose state transition diagram is shown in Fig. 2(b). In this case, the following events make state Sv change to a new state: i) an originating/handoff voice call arrival able to activate de-allocation with rate δdiΛ (2 ≤ i ≤ n) associated with the corresponding new state S15(i); ii) an originating (a handoff) voice call arrival not able to activate de-allocation with rate (δ d0 + δ d1) λ v (( δ d0 + δ d1) λ h ) associated with the corresponding new state S 4 (S 6 ); iii) a data packet arrival with rate λ d associated with the corresponding new state S 8 ; iv) a voice call completion or handoff with rate v µ associated with the corresponding new state S 1; v) channel release by a data user using i (1 ≤ i ≤ n) channels with rate id iµ d associated with the corresponding new state S 12(i). Conversely, the following states can move back to state S v : i) state S 1 with rate Λ caused by an originating/handoff voice call arrival; n ii) state S 8 with rate Σ id iµ d due to channel release i=1 by data users; iii) state S 11(s) (0 ≤ s ≤ 1) with rate (d1 + 1) µd because one channel is released by a data user to let a queued originating/handoff call get service; iv) state S12(i) (1 ≤ i ≤ n) with rate δ piλd because of a data packet arrival; v) state S9 with rate (v + 1) µ because of a call completion or handoff to let the queued data packet get service, vi) state S 14 (i) (2 ≤ i ≤ n) with rate δ diΛ because an originating/handoff voice call arrival gets a channel by gradually de-allocating one of the channels from a data user occupying the most channels; vii) state S4 (S 6) with rate v µ + (q n + 1)η (vµ + (qh + 1) η) due to a voice call completion or handoff to let a queued originating/handoff voice call get service. Again, by equating rates out of state S v to rates into state S v, the balance equation can be obtained and shown as follows:

390


(vµ +

n

Σ

i= 0

δ diΛ +

n

Σ

i=1

id iµ d + λ d )PS v

v, d1, d2, ..., dn, qn, qh, qd – 1

= ΛPS 1 + [vµ + (q n + 1)η]PS 4 + [vµ + (q h + 1)η]PS 6 n

Σ δ piλ d PS i=1

+ (v + 1)µ PS 9 + 1

+

Σ (d 1 + 1)µ d PS s= 0

n

12(i)

+

Σ id iµ d PS i=1

+

Σ δ diΛPS i=2

14(i)

v µ + qh η

(7)

.

v + 1, d1 – 1, d2, ..., dn, qn, qh – 1, qd

v, d1, d2, ..., dn, qn, qh, qd

d1 µd

v, d1, d2, ..., dn, qn, qh, qd + 1

λd

λh

8

n

11(s)

λd

v, d1, d2, ..., dn, qn, qh – 1, qd

v µ + (qh + 1) η v, d1, d2, ..., dn, qn, qh + 1, qd λh

n

Case 3: It stands for the case when, v + Σ id i = C, qn i=1 = qh = 0, 1 ≤ qd ≤ Bd with the state transition diagram shown in Fig. 2(c). Comparing Fig. 2(c) with Fig. 2(b), one can easily know that this case is very similar to Case 2 discussed above since only the following modifications based on Case 2 are required: 1) removing state transitions between state Sv and state S1 (S12(i)), 2) adding a state transition from state Sv to state S7 (S10) n with rate Σ id iµ d (vµ), 3) adding a state transition from i=1 state S7 to state Sv with rate λd. Thus, slightly modifying Eq. (7) results in the following balance equation: (vµ +

n

Σ

i= 0

δ diΛ +

n

Σ

i=1

n

Σ id iµ d PS i=1

+

Σ δ diΛPS i=2

14(i)

v, d1, d2, ..., dn, qn, qh, qd – 1 λd

v, d1, d2, ..., dn, qn – 1, qh, qd

v, d1, d2, ..., dn, qn, qh, qd + 1

v µ + qn η

λd

λv

v + 1, d1 – 1, d2, ..., dn, qn – 1, qh, qd

v, d1, d2, ..., dn, qn, qh, qd

d1 µd


v µ + (qn + 1) η

(d1 + 1) µd

Σ (d 1 + 1)µ d PS s=0

n

+

v, d1, d2, ..., dn, qn + 1, qh, qd

(a) Case 4.

λv

(d1 + 1) µd

v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

v, d1, d2, ..., dn, qn + 1, qh, qd

v – 1, d1+ 1, d2, ..., dn, qn + 1, qh, qd

1

8

λv

v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

id iµ d + λ d )PS v

= λ d PS 7 + [vµ + (q n + 1)η]PS 4 + [vµ + (q h + 1)η]PS 6 + (v + 1)µ PS 9 +

(qn + 1) η

(d1 + 1) µd

11(s)

(b) Case 5.

(8)

. n

Case 4: It is the case when v + Σ id i = C, qn = 0, 1 ≤ i=1 q h ≤ B v, 0 ≤ qd ≤ B d. For this case, all data users in service use exactly one channel; hence, incoming handoff voice calls must be queued since no degradation can be performed. Following a philosophy similar to previous cases, one can obtain the state transition diagram shown in Fig. 3(a) and the balance equation expressed below. (v µ + qhη + Λ + d1µ d + λd )P Sv

v, d1, d2, ..., dn, qn, qh, qd – 1 λd

v, d1, d2, ..., dn, qn – 1, qh, qd

v + 1, d1 – 1, d2, ..., dn, qn, qh – 1, qd

v, d1, d2, ..., dn, qn, qh, qd + 1

qn η

λv

λd

v, d1, d2, ..., dn, qn, qh, qd

d1 µd


(d1 + 1) µd v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

(qn + 1) η v µ + qhη

λh

λv

v, d1, d2, ..., dn, qn, + 1 qh, qd

v, d1, d2, ..., dn, qn, qh – 1, qd

= λ hP S5 + (qn + 1)η PS4 + [v µ + (q h + 1) η ]P S6 + λ dP S7 + (d 1 + 1) λ d P S11(0).

(c) Case 6.

(9)

Fig. 3 State transition diagrams for strategy RAS 1 (Cases 4-6)

n

Case 5: It denotes the case when v + Σ id i = C, 1 ≤ i=1 q n ≤ B v, q h = 0, 0 ≤ q d ≤ B d with the state transition diagram shown in Fig. 3(b). For this case, incoming new voice calls must be queued since no degradation can be performed with the same reasoning as in Case 4. Referring to Fig. 3(b), the balance equation is expressed as follows: (v µ + qnη + Λ + d1µ d + λd )P Sv

(v µ + qnη + qhη + Λ + d1µ d + λd)P Sv

= λ vP S3 + [v µ + (qn + 1) η ]PS 4

= λ vP S3 + (qn + 1) η P S4 + [v µ + (qh + 1) η ]PS 6

+ [v µ + (q h + 1) η ]P S6 + λ dP S7 +

1

Σ

s=0

(d 1 + 1) µ dP S11(s) .

n

Case 6: It depicts the case when v + Σ id i = C, 1 ≤ i=1 qn ≤ Bv, 1 ≤ qh ≤ Bv, 0 ≤ qd ≤ Bd, with the state transition diagram shown in Fig. 3(c) and the balance equation given below. For this case, both incoming new and handoff voice calls are queued for the same reason as in Case 4.

+ λdP S7 + (d 1 + 1) µ dP S11(0) + λh PS 5. (10)

(11)

The above balance equations enable one to


derive steady-state probabilities which subsequently allow one to derive the blocking (forced termination) probability of a new (handoff) voice call, the data packet dropping probability, and delay for each type of call/packet. First, the blocking probability of a new voice call P vb and the forced termination probability of a handoff voice call P ft can be obtained using the following relations (Ferng and Tsai, 2005): P vb = P vbf + (1 – P vbf)Pvbt , P ft = P ftf + (1 – P ftf)P ftt,

where P vbf and P ftf (P vbt and P ftt ) indicate the full (time-out) blocking probability of a new voice call and that of a handoff voice call, respectively. The four probabilities are given as follows: Pvbf = P ftf =

Pvbt =

P ftt =

Σ

Σ

PS v ,

S v ∈ S V1

S v ∈ S V2

q nηPS v

λ n(1 – Pvbf )

Σ

S v ∈ S V2

q hηPS v

λ n(1 – P ftf )

(14)

,

(15)

,

(16)

where sets S V1 and SV2 are defined as follows: SV1 = {Sv|v +

n

Σ id i = C, 0 ≤ v ≤ C, i=0

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, 0 ≤ q n ≤ B v,

0 ≤ qh ≤ Bv, qn + qh = Bv, and 0 ≤ qd ≤ Bd}, SV2 = {Sv|v +

n

Σ id i = C, 0 ≤ v ≤ C, i=0

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, 0 ≤ q n ≤ B v,

0 ≤ qh ≤ Bv, 0 ≤ qn + qh ≤ Bv, and 0 ≤ qd ≤ Bd}. As for the data packet dropping probability P d, it can be expressed as follows: Pd =

Σ

S v ∈ S V3

PS v ,

(17)

where SV3 = {Sv|v +

n

Σ

i=0

id i = C, 0 ≤ v ≤ C,

0 ≤ di ≤ C i

packet D d are shown in the following via application of the famous Little’s formula.

Σ

D nv =

S v ∈ S V2

Σ

D hv =

S v ∈ S V2

for 1 ≤ i ≤ n, 0 ≤ q n ≤ B v,

0 ≤ qh ≤ Bv, 0 ≤ qn + qh ≤ Bv, and qd = Bd}. Finally, the delay of a new voice call D nv , the delay of a handoff voice call D hv, and the delay of a data

S v ∈ S V2

,

(18)

,

(19)

.

(20)

q h PS v

λ h(1 – P ftf )

Σ

Dd =

q n PS v

λ n(1 – Pvbf )

(12) (13)

391

q d PS v

λ d (1 – Pd )

2. Analysis of Strategy RAS 2 To further get performance improvement for the real-time service, strategy RAS2 incorporates preemption into strategy RAS 1. Strategy RAS 2 checks the status of the data buffer when all channels are occupied and no de-allocation is available upon a voice call request arrival. If the data buffer can still accommodate requests, one of the data users (if any) which posses only one channel at this moment is preempted and put into the data buffer to await new service so that the voice call can use the channel previously occupied by the preempted data user. Otherwise, the voice call request is queued in the voice buffer. Note that the state space of strategy RAS 2 is the same as that of strategy RAS 1 described in Eq. (1). Again, six cases like strategy RAS 1 for analyzing strategy RAS 2 are considered. Since balance equations and state transition diagrams of Case 1 for both strategies RAS 1 and RAS2 are the same, this case is neglected and the remaining cases are discussed in the following. For simplicity, only differences for each case between strategies RAS 2 and RAS 1 are emphasized. Case 2. (v +

n

Σ id i = C, qn = 0, qh = 0, qd = 0): i=1

Because of preemption, the following modifications are required for this case as compared to the corresponding case of strategy RAS 1 : (i) Rate from state S v to state S 4 (S 6 ) should be changed to δ d0 λ v ( δ d0 λ h ) instead of ( δ d0 + δ d1) λ v (( δ d0 + δ d1) λ h ) since preemption can not be performed under such a situation and the originating (handoff) voice call request is buffered. (ii) State transition from state Sv to state S9 with rate δ d1Λ should be added to account for preemption when an originating/handoff voice call request arrives. (iii) State transitions from state S11 (s) (s = 0, 1) to S v with rate (d1 + 1) µ d disappear due to the fact that buffering for the originating or handoff call under such a situation is not necessary because preemption is possible. With these modifications, Fig. 4(a) shows the resultant state transition diagram which gives the following balance equation.

392


v – 1, d1, ..., dn – 1 – 1, dn + 1, qn, qh, qd

δ dn Λ

v – 1, d1 – 1, d2 + 1, d3, ..., dn, qn, qh, qd

v – 1, d1, d2, ..., dn, qn, qh, qd

v, d1, d2, ..., dn, qn, qh, qd + 1

λd

δd2 Λ

(d1 + 2d2 + ... + ndn) µd

Λ

δd0 λh

vµ

δpn λ d v, d1, d2, ..., dn – 1, qn, qh, qd

v, d1, d2, ..., dn, qn, qh + 1, qd

v µ + (qh + 1) η δd0 λv

v, d1, d2, ..., dn, qn, qh, qd

v, d1, d2, ..., dn, qn + 1, qh, qd

nd1 µd

v µ + (qn + 1) η δd1Λ

δp1λ d v, d1 – 1, d2, ..., dn, qn, qh, qd

v + 1, d1 – 1, d2, ..., dn, qn, qh, qd + 1 (v + 1)µ

d1 µd

δdn Λ

δd2Λ

v + 1, d1, ..., dn – 1 + 1, dn – 1, qn, qh, qd v + 1, d1 + 1, d2 – 1, d3, ..., dn, qn, qh, qd

(vµ +

(a) Case 2.

δ dn Λ (d1 + 2d2 + ... + ndn) µd

δd1 Λ

v – 1, d1, d2, ..., dn, qn, qh, qd – 1

δd0 λh

vµ v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd δF(d1 + 1)µd

n

v, d1, d2, ..., dn, qn, qh + 1, qd

v µ + (qh + 1) η δd0 λv

v, d1, d2, ..., dn, qn, qh, qd

v, d1, d2, ..., dn, qn + 1, qh, qd v µ + (qn + 1) η δd1Λ

v – 1, d1 + 1, d2, ..., dn, qn + 1, qh, qd δF(d1 + 1)µd (d1 + 2d2 + ... + ndn) µd v, d1, d2, ..., dn, qn, qh, qd – 1

v + 1, d1 – 1, d2, ..., dn, qn, qh, qd + 1

λd δdn Λ

v + 1, d1, ..., dn – 1 + 1, dn – 1, qn, qh, qd

δd2Λ

(v + 1)µ

v + 1, d1 + 1, d2 – 1, d3, ..., dn, qn, qh, qd

(b) Case 3.

v, d1, d2, ..., dn, qn, qh, qd – 1

λd v, d1, d2, ..., dn, qn, qh – 1, qd

v µ + qh η

v, d1, d2, ..., dn, qn, qh, qd + 1

λd

λh

v µ + (qh + 1) η v, d1, d2, ..., dn, qn, qh, qd

v + 1, d1 – 1, d2, ..., dn, qn, qh – 1, qd

λh

δFd1 µ d

δF(d1 + 1)µd

v, d1, d2, ..., dn, qn, qh + 1, qd

λv v, d1, d2, ..., dn, qn + 1, qh, qd

(c) Case 4. Fig. 4

Some state transition diagrams for strategy RAS 2 (Cases 2-4)

(vµ +

n

n

Σ δ diΛ + iΣ= 1 id iµ d + λ d)PS i= 0

v

= ΛPS 1 + [vµ + (q n + 1)η]PS 4 + [vµ + (q h + 1)η]PS 6 + (v + 1)µ PS 9 + n

+

Σ

i=2

Case 3. (v +

n

Σ δ piλ d PS i=1

δ diΛPS 14(i) . n

Σ

i=1

n

12(i)

v

+

Σ id iµ d PS i=1

+

Σ

i=2

Case 4. (v +

δ diΛPS 14(i) +

1

Σ

s=0

n

Σ

i=1

id iµ d PS 8

δ F (d 1 + 1)µ d PS 11(s) . (22)

n

Σ id i = C, qn = 0, 1 ≤ qh ≤ Bv, 0 ≤ qd ≤ Bd): i=1

Compared to the corresponding case of strategy RAS 1 , this case needs the following modifications: (i) Rate for the state transition from state S v to state (v + 1, d 1 – 1, d 2 , ... , d n , q n , q h – 1, q d ) should be changed from d 1µ d to δ Fd 1µ d to denote that existence of this state transition is valid only when the data buffer is full. (ii) Rate for the state transition from state S 11 (0) to state S v should be changed from (d 1 + 1)µ d to δ F(d 1 + 1)µ d to denote when the state transition is valid. Again, the resultant state transition diagram after suitable modifications is shown in Fig. 4(c) and the balance equation is expressed as follows: (v µ + qhη + Λ + δ Fd1µ d + λd )P Sv

(qn + 1)η

v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

n

+ (v + 1)µ PS 9 + δ d1ΛPS 10 +

v, d1, d2, ..., dn, qn, qh, qd + 1

λd

δd2 Λ

n

Σ δ diΛ + iΣ= 1 id iµ d + λ d)PS i= 0

= λ d PS 7 + [vµ + (q n + 1)η]PS 4 + [vµ + (q h + 1)η]PS 6

v – 1, d1, ..., dn – 1 – 1, dn + 1, qn, qh, qd v – 1, d1 – 1, d2 + 1, d3, ..., dn, qn, qh, qd

for this case as compared to the corresponding case of strategy RAS1 because of preemption. In addition, the rate for the state transition from state S 11(s) (s = 1, 2) to Sv should be changed from (d1 + 1)µd to δ F(d1 + 1) µ d to denote that existence of these state transitions is valid only when the data buffer is full and the state transition from state S 10 to S v with rate δ d1 Λ should be added to depict preemption on data users. Therefore, the resultant state transition diagram after suitable modifications is shown in Fig. 4(b) and the balance equation is given as follows:

8

(21)

id i = C, qn = 0, q h = 0, 1 ≤ q d ≤ B d):

For the same reasoning in the previous case, the modifications in (i)-(ii) of that case are still required

= λ hP S5 + (qn + 1)η P S4 + [v µ + (q h + 1) η]P S6 + λd PS 7 + δ F(d1 + 1) µ dP S11(0).

(23)

Following a similar philosophy in case 4, one can further derive balance equations for the remaining cases, i.e., cases 5 and 6 with special attention to existence of some state transitions, including state transitions from state S11(s) (s = 0,1) to state Sv, state transition from state Sv to state (v + 1, d 1 – 1, d 2, ... , dn, qn – 1, qh, qd), and state transition from state Sv to state (v + 1, d 1 – 1, d 2 , ... , d n , q n , q h – 1, q d ). For brevity, these two cases are omitted here. Finally, Eqs. (12)-(13) and (17)-(20) are still applicable for this case to get the blocking probability of a new voice call, the forced termination probability of a handoff voice call, the data packet dropping probability, the delay time for a new voice call, the delay time for a handoff voice call, and the delay time for a data packet, respectively.


To have a compact shape for this paper, we shall not further elaborate on analyses for strategies RASth1 and RASth2. In stead, we sketch approaches to get analytic results for strategies RASth1 and RASth2 in the Appendix.

393

W d are 4.2% and 5.9%, respectively). Compared to strategy RAS 1 , strategy RAS 2 gets 23%, 28%, 21%, and 26% of improvement in P vb , P ft , D nv , and D hv, respectively, as well as 0.36% and 0.32% of degradation in P d and D d, respectively, when µ d = 220 µ v.

IV. NUMERICAL RESULTS AND DISCUSSIONS 2. Effects of Data Packet Transmission Rates To obtain related probabilities and performance measures, a recursive procedure for the analytic calculation is employed similar to (Ferng and Tsai, 2005). Applying the analytic models along with simulation experiments run on IBM compatible PCs with C codes, we study system performance for GPRS, including blocking probability of a new voice call P vb , forced termination probability of a handoff voice call P ft , data packet dropping probability P d , and delay time for a new voice call Dnv, delay time for a handoff voice call D hv, and delay time for a data packet D d. Aside from the aforementioned strategies, two extra strategies, i.e., RAS bp which merely employs priorities among buffers like schemes in (Lin and Lin, 2001), (Lin, 2003), and RAS hp which applies both priorities among buffers and service priorities analogous to schemes in (Ferng and Tsai, 2005), are included for comparison. Concerning the effect of burstiness of data traffic and packet retransmission rate RR caused by preemption, simulations are conducted to get the corresponding results as well. Let us now discuss the results from the following six aspects. For simplicity, we set µ v = 1 for all numerical examples.

Shown in Figs. 6(a)-(c) are the effects of data packet transmission rates ( µ d ) on the blocking probability of a new voice call, forced termination probability of a handoff voice call, and data packet dropping probability, respectively. Note that a larger value of µd implies a shorter packet transmission time (packet size). Therefore, performance goes down as µd increases. Regarding the sensitivity to data packet transmission rates with respect to the blocking probability of a new voice call and the forced termination probability of a handoff voice call, strategy RAS2 is less sensitive than strategy RAS1 which is in turn less sensitive than strategies RASbp and RAShp (see Figs. 6(a)-(b)). Compared to strategies RAS bp, RAS hp and RAS1 under µd = 50 µv (µd = 100µv), strategy RAS2 gets 94.4% (23.9%), 108% (36.4%), and 48.7% (15.1%) of improvement in P vb as well as 170% (78.3%), 139% (46.3%), and 55.6% (16%) of improvement in P ft. However, comparable results are exhibited by strategies RAS1, RAS2, RASbp, and RAS hp as shown in Fig. 6(c). As for the effect of data packet transmission rates over delay times, the results are similar to the corresponding results over probabilities and are omitted here for brevity.

1. Effects of Data Arrival Rates 3. Effects of the Data Buffer Size Under various data arrival rates, performance of strategies RAS 1, RAS 2, RAS bp, RAS hp is observed and shown in Figs. 5(a)-(f). First, one may note that strategy RASbp does not differentiate QoS received by new and handoff voice calls, e.g., P vb = P ft. This seems to be not acceptable from the viewpoint of users. Therefore, strategy RAS hp further employs service priorities to get better performance for handoff voice calls but it degrades performance of new voice calls (see Figs. 5 (a)-(b), 5(d)-(e)) without affecting performance of data packets (see Figs. 5(c) and 5(f)). Second, applying de-allocation to strategy RAS1 improves performance of both new and handoff voice calls with negligible degradation of performance for data packets as compared to strategy RAShp. The down shifting phenomenon is explicitly illustrated by crossings between strategies RASbp and RAS1 in Figs. 5(a) and 5(d). Finally, preemption on data packets utilized by strategy RAS2 reduces significantly the effect of data arrival rates and lets strategy RAS 2 have the best performance for new and handoff voice calls among the four strategies with a bit of degradation in performance for data packets (the maximum observed increments in Pd and

In Figs. 7(a)-(b), two different trends are shown when observing Pvb and Pft vs. the data buffer size Bd for strategies RAS1 and RAS2: smooth increase for strategy RAS1 and smooth decrease for strategy RAS 2. The reasons why different trends are exhibited by strategies RAS 1 and RAS2 are given as follows. With more data buffer space, more incoming (preempted) data packets are allowed to be buffered for strategy RAS 1 (RAS2), causing originating/handoff voice calls to have lower (higher) chances to enter the system. Obviously, larger Bd leads to a smaller data packet dropping probability but a higher packet delay proability as shown in Figs. 7(c)-(d) in which small but noticeable differences between strategies RAS1 and RAS2 are observed (higher values for strategy RAS2 with higher percentage of difference as the data buffer size increases because of more frequent preemption). 4. Effects of the Minimum Number of Channels Prohibiting Preemption Associated with Data Users We may define a minimum number of channels

394


× 10–3

2.2 2.1

Pvb

2 1.9

RASbp RASbp RAShp RAShp RAS1 RAS1 RAS2 RAS2

2.4

Line: Analysis Dot: Simulation

× 10–3 RASbp RASbp RAShp RAShp RAS1 RAS1 RAS2 RAS2

2.2 2

Pft

2.3

1.8

1.8 1.6

1.7


1.4

1.6 1.2

1.5

1

1.4 1.3 40

60

80

100

120 140 λd

160

180

200

0.8

220

(a) Blocking probability of a new voice call

0.035 0.03

Pd

0.025


60

80

100

120 140 λd

160

180

200

220

(b) Forced termination probability of a handoff voice call 4.6


× 10–3

4.4 4.2 4 3.8

Dnv

0.04

40

0.02

3.6



3.4

0.015

3.2

0.01

3 0.005 0 40

2.8 60

80

100

120 140 λd

160

180

200

2.6 40

220

60

(c) Data packet dropping probability × 10–3 RASbp RASbp 4.5 RAShp RAShp RAS1 4 RAS1 RAS2 3.5 RAS2

100

120 140 λd

160

180

200

220

200

220

(d) Delay time of a new voice call

5

× 10–3 RASbp RASbp 12 RAShp RAShp 11 RAS1 RAS1 10 RAS2 RAS2 9 13


Dd

Dhv

80


3 8 2.5

7

2 1.5 40

6 60

80

100

120 140 λd

160

180

200

220

(e) Delay time of a handoff voice call Fig 5

5 40

60

80

100

120 140 λd

160

180

(f) Delay time of a data packet

Performance comparison among strategies RAS 1, RAS 2, RAS bp, and RAS hp under various data packet arrival rates with C = 7, η = 0.5 µv, µ d = 90µ v, λ v = 2.6µ v, n = 3, B v = 7, and B d = 21

prohibiting preemption np associated with data users. When the number of channels occupied by each data users is lower than or equal to n p , preemption is not

allowed anymore. Originally, we set n p =1 for the proposed strategies. In the following, we also set n p = 3 to observe the effects caused by different values


3.2

× 10–3

difference (%) = RASbp RASbp RAShp RAShp RAS1 RAS1 RAS2 RAS2


3 2.8 2.6

Pvb

2.4 2.2 2 1.8 1.6 1.4 1.2 50

55

60

65

70

75 µd

80

85

90

95

100

(a) Blocking probability of a new voice call 3

Obviously, much lower P vb, P ft, Dnv , and D hv are obtained for np = 1 because more channels are available for voice calls as compared to those with np = 3. The cost one may pay to gain this advantage is that performance of P d and D d degrades a little. Note that percentages of improvement shrink when switching λ v from 2 to 3 because the fact of more voice calls but fewer data users entering the system when switching λv from 2 to 3 makes preemption dominate less in performance improvement for voice calls. 5. Effects of Data Traffic Modeling


2.5

Pft

2

1.5

1

0.5 50

55

60

65

70

75 µd

80

85

90

95

100

(b) Forced termination probability of a handoff voice call 0.045



0.04 0.035 0.03 Pd

the result for n p = 3 – the result for n p = 1 × 100 . the result for n p = 1

× 10–3 Line: Analysis Dot: Simulation

0.025 0.02 0.015 0.01 50

55

60

65

70

75 µd

80

85

90

95

100

(c) Data packet dropping probability Fig. 6

395

Performance comparison among strategies RAS 1, RAS 2 , RAS bp, and RAShp under various data packet transmission rates with C = 7, λ v = 2.6 µ v, η = 0.5 µ v, λ d = 140 µv, n = 3, B v = 7, and B d = 21

of n p. The results are shown in Table 1 in which the difference in percentage is obtained via

Poisson processes are popular and simple in modeling traffic patterns. However, using Poisson to model data traffic may fail to capture the nature of data traffic, e.g., self-similarity (Leland et al., 1994), (Paxson and Floyd, 1995). Noting that exponentially distributed inter-arrival times are exhibited for Poisson Processes, we also adopt the Pareto distribution which is a heavytailed distribution with infinite variance to model inter-arrival times for data traffic to investigate the effects of different traffic models. The Pareto distribution with shape parameter γ and location parameter β has the following cumulative distribution function F(t) and probability density function f(t) with mean γβ /( γ –1): F(t) = 1 – ( β/t)γ, γ, β ≥ 0, t ≥ β, f (t) =

γβ γ . tr + 1

For the Pareto distribution, we can associate a Hurst parameter H = (3 – γ )/2 (Ferng and Chang, 2001), (Paxson and Floyd, 1995) (0.5 ≤ H < 1) used to describe the degree of self-similarity or long-range dependence with it. Note that more bursty traffic has a larger value of H. Taking strategy RAS2 as an example, Figs. 8(a)-(b) show the effects of different data traffic models (Poisson and Pareto) on the blocking probability of a new voice call and data packet dropping probability. Here, the result of the forced termination probability of a handoff voice call is omitted because it exhibits similar phenomena resembling the result of the blocking probability of a new voice call. Likewise, delay times are omitted because phenomena analogous to probability-related performance are observed. Obviously, P vb and P d increase as λ v increases as shown in Figs. 8(a)-(b). Furthermore, Fig. 8(a) shows that data traffic modeling and values of

396


Table 1 Effects of the minimum number of channels prohibiting preemption associated with data users observed under strategy RAS 1 with or λ v = 2µ v or 3µ v, λ d = 140µ v, µ d = 90µ v, η = 0.5µ v, n = 4, Bv = 7, and Bd = 21 λv = 2µ v (n p = 3) λv = 2µ v (n p = 1) Difference (%) λv = 3µ v (n p = 3) λv = 3µ v (n p = 1) Difference (%)

1.6

P vb

Pft

Pd

D nv

D hv

Dd

0.000546 0.000413 +32.20 0.003504 0.003284 +6.69

0.000437 0.000303 +44.22 0.002401 0.002147 +11.83

0.003802 0.003769 +0.88 0.030135 0.030387 -0.829

0.001084 0.000795 +36.35 0.006988 0.006501 +7.49

0.000905 0.000619 +46.21 0.0048 0.004237 +13.29

0.003552 0.003575 -0.59 0.015685 0.015818 -0.84

× 10–3

1.2

1.55


1.15

RAS1 RAS1 RAS2 RAS2

1.1


Pft

Pvb

1.5

× 10–3

1.45

1.05

1.4

11

1.35

0.95

1.3 10

20

15

0.9 10

25

25

Bd

Bd (a) Blocking probability of a new voice call 0.024 Line: Analysis Dot: Simulation

0.023

20

15

RAS1 RAS1 RAS2 RAS2

0.022

(b) Forced termination probability of a handoff voice call × 10–3 RAS1 RAS1 RAS2 9 RAS2

10

RAS1 RAS1 RAS2 RAS2


8

0.02

Dd

Pd

0.021

0.019

7

0.018 6

0.017 0.016

5

0.015 0.014 10

15

20

25

Bd (c) Data packet dropping probability

4 10

20

15

25

Bd (d) Delay time of a data packet

Fig. 7 Effects of data buffer sizes under C = 7, λv = 2.6µ v , η = 0.5 µv , λ d = 140 µ v , n = 3 and B v = 7

the Hurst parameter have little effect on voice call performance because the voice buffer has a higher priority than the data buffer for strategy RAS 2 . However, data packet dropping probabilities deviate upwardly and profoundly from the result of the Poisson data traffic for data traffic with high Hurst parameters.

6. Investigation of Data Packet Retransmission Caused by Preemption In strategy RAS2, preemption causes data packets to halt service and wait for retransmission in the buffer. What percentage of packets may be preempted? Fig. 9 answers this question by showing the data packet


3

× 10–3

1.4

Poission arrival H = 0.6 H = 0.7 H = 0.8

2.5

397

λ v = 2.6 µv

1.2

λ v = 4.0 µv

1

RR (%)

Pvb

2 1.5 1

0.6 0.4

0.5 0

0.8

0.2

2

2.1

2.2

2.3 2.4

2.5 λv

2.6

2.7

2.8

2.9

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

3

λd

(a) Blocking probability of a new voice call Fig. 9

0.045 Poission arrival H = 0.6 H = 0.7 H = 0.8

0.04 0.035

Pd

0.03 0.025 0.02 0.015 0.01 0.005 0

2

2.1

2.2

2.3 2.4

2.5 2.6 λv

2.7

2.8

2.9

3

(b) Data packet dropping probability Fig. 8

Effects of data traffic burstiness over strategy RAS2 under various arrival rates of new voice calls with C = 7, η = 0.5 µv, λ d = 140 µv, µ d = 90 µ v, n = 3, B v – 7, and B d = 21

retransmission rate (RR). First, the maximum observable value of RR shown in Fig. 9 less than 1.3%, i.e., only a small portion of packets are retransmitted caused by preemption. Second, we is see that more voice arrivals (say, λ v = 4.0µ v) result in higher frequency of preemption, thus higher RR as compared to the case of fewer voice arrivals (say, λv = 2.6µv). Third, RR first goes down as λ d increases because of less available data buffer space reduces occurrence of preemption; then it slightly goes up beyond a certain point since more data packets entering the system with very short service times activate preemption upon departure of some packets (Note that this only has a small effect on RR since it occurs when the data buffer is near full). 7. Performance Comparison Among Strategies RAS 1, RAS 2 , RAS th1 , and RAS th2 Figures 10(b)-(f) explicitly show that threshold

Data packet retransmission rates for strategy RAS 2 under various data packet arrival rates with C = 7, λ v = 2.6 µ v , and 4.0µv, h = 0.5µv , µd = 90µv, n = 3, B v = 7, and B d = 21

control on the voice buffer makes P ft , P d , D nv , D hv, and Dd perform even better. In particular, Pft and D nv. For Pft, Pd, Dhv, and Dd, the improvement comes from fewer new incoming voice calls which definitely cause fewer delays for new voice calls, i.e., D nv . Among the four strategies, strategy RAS th2 performs best in terms of these five performance metrics. However, threshold control on the voice buffer deteriorates performance in P vb (see Fig. 10(a)) since less buffer space is arranged for new voice calls. Further distinguishing performance for strategies RAS th1 and RASth2, one can see that strategy RASth2 still performs a bit better than strategy RAS th1 . If P vb still falls within the acceptable range, then strategy RAS th2 offers the best other performance metrics. From the viewpoint of service differentiation and QoS guarantee, strategy RAS th2 is most suggested. Finally, both strategies RAS 2 and RAS th2 offer much better performance for voice calls than other related strategies proposed in (Ferng and Tsai, 2005), (Lin and Lin, 2001), and (Lin, 2003), (as well as (Chen et al., 2003)) with negligible performance degradation for data packets based on the above results and discussions. This implies that both strategies RAS2 and RAS th2 deserve to be employed in the GPRS system. Of course, strategy RASth2 is even better than strategy RAS 2 if the performance of P vb can be maintained within an acceptable range. V. CONCLUSIONS Using a comprehensive approach and utilizing some promising techniques, e.g., priority, de-allocation, and preemption, we have designed four resource allocation strategies. The main design ideas behind the four resource allocation strategies are i) to have less

398


0.045 0.04 0.035

Pvb

0.03

0.018

RAS1 RAS1 RASth1 RASth1 RAS2 RAS2 RASth2 RASth2



0.016 0.014 0.012 Pft

0.05

0.025 0.02


0.01 0.008

0.015 0.006 0.01 0.004

0.005 0 3.2

3.4

3.6

3.8

4

λv

4.2

4.4

4.6

4.8

0.002 3.2

5

(a) Blocking probability of a new voice call

0.3

Pd

0.25 0.2



0.06 0.05 0.04 0.03

0.1

0.02

0.05

0.01

3.4

3.6

3.8

4

λv

4.2

4.4

4.6

4.8

0 3.2

5


3.4

(c) Data packet dropping probability

0.015

λv

4.2

4.4

4.6

4.8

5


3.6

3.8

4

λv

4.2

4.4

4.6

4.8

5

4.8

5

0.09



0.08 0.07 0.06 Dd

Dhv

0.02

4

(d) Delay time of a new voice call

0.03 0.025

3.8

(b) Forced termination probability of a handoff voice call

0.15

0 3.2

3.6

0.07

Dnv

0.35

3.4

0.05



0.04

0.01

0.03 0.005 0 3.2

0.02 3.4

3.6

3.8

4

λv

4.2

4.4

4.6

4.8

(e) Data time of a handoff voice call

5

0.01 3.2

3.4

3.6

3.8

4

λv

4.2

4.4

4.6

(f) Delay time of a data packet

Fig. 10 Performance comparison among strategies RAS 1, RAS 2, RASth1, and RAS th2 under various arrival rates of new voice calls with C = 7, η = 0.5µv, λ d = 140 µ v, µ d = 90µv , n = 3, B v = 7, T = 3, and B d = 21

blocking probability of a new voice call, forced termination probability of a handoff voice call, and data packet dropping probability by using buffering; ii) to get even better performance for voice calls with

the cost of a bit of degradation on data packets by using priority, de-allocation, and preemption; iii) to gain superior QoS differentiation between new and handoff voice calls by using priority and threshold


control on the voice buffer. To examine the effectiveness of the four proposed strategies, both analytic and simulation approaches are employed. For the former, the Markov chain is applied to obtain general analytic results. Through extensive numerical experiments, the following observations have been made. (1) The four proposed strategies can get even better performance for new and handoff voice calls with a little performance degradation for data packets as compared to strategies RASbp following design ideas in (Lin and Lin, 2001), (Lin, 2003), and RAS hp designed according to ideas in (Ferng and Tsai, 2005) observed under different data packet arrival rates, data packet transmission rates, and data buffer sizes. (2) The (conditional) preemption utilized in strategies in RAS2 and RASth2 only results in a very small data packet retransmission rate (not higher than 1.3% based on previous numerical examples) to get better performance for voice calls. (3) Inclusion of threshold control on the voice buffer into strategies RASth1 and RASth2 provides easy and feasible QoS differentiation between new and voice calls as well as performance improvement for data packets. Based on the numerical results and observations, we suggest strategies RAS2 and RAS th2 for GPRS, in particular, strategy RAS th2. Of couse, the theoretical performance analysis for these strategies given in this paper also affords researchers useful performance evaluation tools in the system design stage. NOMENCLATURE Bv Bd C di D nv D hv Dd P vb P ft

voice buffer size data buffer size the number of radio channels the number of data packets using i channels delay time for a new voice call delay time for a handoff voice call delay time for a data packet blocking probability of a new voice call forced termination probability of a handoff voice call Pd data packet dropping probability P vbf (P ftf) full blocking probability of a new voice call (handoff voice call) P vbt (P ftt) time-out blocking probability of a new voice call (handoff voice call) qn the number of new voice calls in queue qh the number of handoff voice calls in queue qd the number of data packets in queue RR the data packet retransmission rate T the maximum number of queued new voice calls v the number of voice calls in service λd rate of data packets λh handoff rate of voice calls

λv Λ 1/η 1/ µ d 1/ µ v 1/ µ

399

rate of new voice calls the total voice arrival rate dwelling time of a voice call packet transmission time voice call holding time occupancy time of a voice call ACKNOWLEDGEMENT

The second author would like to thank the financial support from the National Science Council, Taiwan, R.O.C. under the following two contracts: NSC 952221-E-011-029 and NSC 96-2221-E-011-020-MY3. REFERENCE Chen, W. Y., Wu, J. L. C., and Lu, L. L., 2003, “Performance Comparisons of Dynamic Resource Allocation with/without Channel De-Allocation in GSM/GPRS Networks,” IEEE Communications Letters, Vol. 7, No. 1, pp. 10-12. Dahlman, E., Gudmundson, B., Nilsson, M., and Skold, J., 1998, “UMTS/IMT-2000 Based on Wideband CDMA,” IEEE Communications Magazine, Vol. 36, No. 9, pp. 70-80. Eberspacher, J., Vogel, H. J., and Bettstetter, C., 2001, GSM – Switching, Services, and Protocols, Wiley, New York, USA. ETSI, 1999, “Digital cellular telecommunications system (Phase 2+): General Packet Radio Service (GPRS); Overall description of the GPRS radio interface; Stage 2 (GSM 03.64 ver. 7.0.0 Release 1999),” ETSI/TC, Technical Report GSM 03.64. ETSI, 2002, “Universal Mobile Telecommunications System (UMTS): Services and Service Capabilities (3GPP TS 22.105 ver. 5.2.0 Release 5),” ETSI/TS, Technical Report 3GPP TS 22.105. ETSI, 2003a, “Universal Mobile Telecommunications System (UMTS): Service Aspects; Service Principles (3GPP TS 22.101 ver. 5.9.0 Release 5),” ETSI/TS, Technical Report 3GPP TS 22.101. ETSI, 2003b, “Universal Mobile Telecommunications System (UMTS): Quality of Service (QoS) Concept and Architecture (3GPP TS 23.107 ver. 5.9. 0 Release 5),” ETSI/TS, Technical Report 3GPP TS 23.107. Ferng, H. W., and Chang, J. F., 2001, “Characterization of the output of an ATM output Buffer Receiving Self-Similar Traffic,” IEEE Global Communications Conference, Vol. 2, No. 2, pp. 2650-2653. Ferng, H. W., and Tsai, Y. C., 2005, “Using Priority, Buffering, Threshold Control and Reservation Techniques to Improve Channel Allocation Schemes for the GPRS System,” IEEE Transactions on Vehicular Technology, Vol. 54, No. 1,

400


pp. 286-306. Garay, J. A., and Gopal, I. S., 1992, “Call Preemption in Communication Networks,” IEEE Conference on Computer Communications, Vo. 3, pp. 1043-1050, Holma, H., and Toskala, A., 2002, WCDMA for UMTS, Radio Access for Third Generation Mobile Communications, Wiley, New York, USA. Hong, D., and Rappaport, S. S., 1986, “Traffic Model and Performance Analysis for Cellular Mobile Radio telephone System with Prioritized and Nonprioritized Handoff Procedure,” IEEE Transactions on Vehicular Technology, Vol. 35, No. 3, pp. 77-92. Leland, W. E., Taqqu, M. S., Willinger W., and Wilson, D. V., 1994, “On the Self-Similar Nature of Ethernet Traffic (extended version),” IEEE/ACM Transactions on Networking, Vol. 2, No. 1, pp. 1-15. Lin, Y. B., Mohan, S., and Noerpel, A., 1994, “Queueing Priority Channel Assignment Strategies for PCS Hand-Off and Initial Access,” IEEE Transactions on Vehicular Technology, Vol. 43, No. 3, pp. 704-712. Lin, P., and Lin, Y. B., 2001, “Channel Allocation for GPRS,” IEEE Transactions on Vehicular Technology, Vol. 50, No. 2, pp. 375-387. Lin, P., 2003, “Channel Allocation for GPRS with Buffering Mechanisms,” Wireless Networks, Vol. 9, No. 5, pp. 431-441. Liu, H. H., Wu, J. L. C., and Hsieh, W. C., 2002, “Delay Analysis of Integrated Voice and Data Service for GPRS,” IEEE Communications Letters, Vol. 6, No. 8, pp. 319-321. Oh, S. H., and Tcha, D. W., 1992, “Prioritized Channel Assignment in a Cellular Radio Network,” IEEE Transactions on Communications, Vol. 40, No. 7, pp. 1259-1269. Paxson, V., and Floyd, S., 1995, “Wide Area Traffic: The Failure of Poisson Modeling,” IEEE/ACM Transactions on Networking, Vol. 3, No. 3, pp. 226-244. Sollenberger, N. R., Seshadri, N., and Cox, R., 1999, “The Evolution of IS-136 TDMA for Third-Generation Wireless Services,” IEEE Personal Communications, Vol. 6, No. 3, pp. 8-18. Wang, J., Zeng, Q. A., and Agrawal, D. P., 2003, “Performance Analysis of a Preemptive and Priority Reservation Handoff Scheme for Integrated Service-Based Wireless Mobile Networks,” IEEE Transactions on Mobile Computing, Vol. 2, No. 1, pp. 65-75. Zhang, Y., and Soong, B. H., 2004, “Performance Evaluation of GSM/GPRS Networks with Channel Re-Allocation Scheme,” IEEE Communications Letters, Vol. 8, No. 5, pp. 280-282. Zheng, J., and Regentova, E., 2004, “Performance Analysis of Channel De-Allocation Schemes for

Dynamic Resource Allocation in GSM/GPRS Networks,” IEE Electronics Letters, Vol. 40, No. 24, pp. 1544-1545. Manuscript Received: Aug. 31, 2007 Revision Received: Jan. 25, 2008 and Accepted: Feb. 25, 2008 APPENDIX ANALYSIS OF STRATEGIES RASth1 and RAS th2 The difference between strategies RASth1 and RAS1 is buffering control. For strategy RASth1, an extra threshold control over originating voice requests is included, i.e., the maximum number of originating voice requests in the buffer is fixed to T. Slightly modifying state space of strategy RAS 1 then leads to state space of strategy RAS th1, which is described as follows: SRASth1 = {Sv|0 ≤ v +

n

Σ id i ≤ C, 0 ≤ v ≤ C, i=1

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, i = 1, ..., n,

0 ≤ qn ≤ T, 0 ≤ qh ≤ B v, 0 ≤ qd ≤ Bd}. (24) All cases when analyzing strategy RASth1 can be easily derived following a similar philosophy for stratn egy RAS 1 except for one special case, i.e., v + Σ id i i=1 = C, q n = T, 1 ≤ q h ≤ B v – T, 0 ≤ q d ≤ B d . For compactness, we only discuss this special case in the following. In Fig. 11, the corresponding state-transition diagram is given. Based on the state transition diagram, the balance equation is derived as follows: (v µ + qhη + qvη + λ h + d 1µ d + λd )P Sv = λ hP S5 + λ vP S3 + [vµ + (qh + 1) η ]P S6 + λdR S7 + (d1 + 1) µd PS 11(0) .

(25)

Although Eqs. (12) and (13) are still applicable to calculate the blocking probability of a new voice call and the forced termination probability of a handoff voice call for strategy RAS th1 , calculation of Pvbf, Pftf, Pvbt, and Pftt should follow the following four equations. Pvbf =

P ftf =

Pvbt =

Σ

PS v +

Σ

PS v ,

Σ

q vηPS v

S v ∈ S V4

S v ∈ S V6

S v ∈ S V7

Σ

S v ∈ S V5

Λ(1 – Pvbf )

PS v ,

(26) (27)

,

(28)


v, d1, d2, ..., dn, qn, qh, qd – 1

SV7 = {Sv|v +

λd v, d1, d2, ..., dn, qn, qh – 1, qd

v µ + qh η

v, d1, d2, ..., dn, qn, qh, qd + 1

λd

λh qn η v, d1, d2, ..., dn, qn – 1, qh, qd

v, d1, d2, ..., dn, qn, qh, qd

λv d1µd

v – 1, d1 + 1, d2, ..., dn, qn, qh + 1, qd

for 1 ≤ i ≤ n, 0 ≤ qn < T,

0 ≤ qh ≤ Bv – T, and 0 ≤ q d ≤ Bd},

λh (d1 + 1) µd

n

Σ id i = C, 0 ≤ v ≤ C, i=0

0 ≤ di ≤ C i

v µ + (qh + 1) η v, d1, d2, ..., dn, qn, qh + 1, qd

401

SV8 = {Sv|v +

v + 1, d1 – 1, d2, ..., dn, qn, qh – 1, qd

n

Σ id i = C, 0 ≤ v ≤ C, i=0

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, 0 ≤ qn < T,

0 ≤ qh ≤ Bv, and 0 ≤ qd ≤ B d},

Fig. 11 The state transition diagram for strategy RAS th1 when v + n Σ id i = C, qn = T, 1 ≤ qh ≤ Bv – T, and 0 ≤ qd ≤ Bd i=1

P ftt =

Σ

S v ∈ S V8

q hηPS v

Λ(1 – P ftf )

The data packet dropping probability P d is obtained via (29)

.

where

Pd =

Σ

S v∈ S V9

(30)

PS v ,

where

SV4 = {Sv|v +

n

Σ

i=0

id i = C, 0 ≤ v ≤ C,

0 ≤ di ≤ C i

for i ≤ 1 ≤ n, qn = T,

0 ≤ qh ≤ B v – T, and 0 ≤ qd ≤ Bd }, SV5 = {Sv|v +

n

Σ

i=0

id i = C, 0 ≤ v ≤ C,

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, 0 ≤ qn < T,

0 ≤ qh ≤ Bv, qn + qh = Bv, and 0 ≤ qd ≤ Bd}, SV6 = {Sv|v +

n

Σ

i=0

id i = C, 0 ≤ v ≤ C,

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, 0 ≤ qn < T,

0 ≤ qh ≤ Bv, qn + qh = Bv, and 0 ≤ qd ≤ Bd},

SV9 = {Sv|v +

n

Σ

i=0

id i = C, 0 ≤ v ≤ C,

0 ≤ di ≤ C i

for 1 ≤ i ≤ n, 0 ≤ qn < T,

0 ≤ qh ≤ Bv, and qd = B d}. Finally, D nv, D hv, and D d for strategy RASth1 are also obtained via Eqs. (18), (19), and (20), respectively with the following changes: (i) Sv ∈ SV2 in Eq. (18) should be changed to S v ∈ S V7 (ii) S v ∈ S V2 in Eqs. (19) and (20) should be changed to Sv ∈ S V8. As for strategy RAS th2 which is formed by further incorporating the threshold control technique into strategy RAS 2 , its performance measures can be derived using a manner similar to that for strategy RAS th1 . To make the paper compact, the details are omitted and left for interested readers.