Heavy Traffic Limits in a Wireless Queueing Model with Long Range Dependence

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1 Heavy raffic Limits in a Wireless Queueing Model with Long Range Dependence Robert. Buche, Arka Ghosh, Vladas Pipiras Abstract High-speed wireless networks carrying multimedia applications are becoming a reality and the transmitted data exhibit long range dependence and heavy-tailed properties. We consider the heavy traffic approach in working towards queue models under these properties, extending the model in 3] from the short range dependence and light-tailed case. Our focus is on the scalings used in the heavy traffic approach which are determined by combinations of the source rate of an infinite source Poisson model of the arrival process, the tail distribution of data transmitted by these sources, and the rate of variation of the random process (channel process) modeling the wireless medium. A fundamental inequality between the exponent in the power tail distribution of the data from the source and the rate of channel variations is obtain. his inequality is important in both the fast growth and slow growth regimes for the arrival process and along with the source rate is used to define the possible cases for obtaining limit models for the queueing process. Across the cases, the possible limit models include reflected Brownian motion, reflected stable Lévy motion, or reflected fractional Brownian motion. I. INRODUCION he usefulness of the heavy traffic method for wireline systems under short range dependence (SRD) assumptions (for example, Poisson interarrival times of packets) and light tailed (L) assumptions (for example, finite second moments on the packet size and service times) is well established, having a large literature (see 17], 9] and the references therein). Loosely speaking, an actual stochastic queueing system may be intractable for analysis and the heavy traffic method scales time and the state space yielding an asymptotic model giving the queue-size dynamics which retains the important features of the actual queueing system. In the SRD/L case, the limit model is given by reflected Brownian motion (RBM). With the prevalence of wireless systems, the heavy traffic method has been extended to wireless models (for example, 3], 16]). he maor difference from the wireline model is the random environment for transmission due to physical surroundings and mobility characteristics of the user. But like the wireline case, the limit queue models are given by RBM. Recently, there has been a maor paradigm shift from short range dependence and light-tailed to long range dependent (LRD) and heavy-tailed (H) stochastic models of data traffic. LRD stands for a very strong temporal dependence in a time series, and H describes the tails of a Supported by the National Science Foundation grants DMS (RB), DMS (AG), DMS (VP). R. Buche is with the Mathematics Department, NC State Univ., Raleigh, NC 27695, rtbuche@unity.ncsu.edu A. Ghosh is with the Statistics Department, Iowa State Univ., Ames, IA 511, apghosh@iastate.edu V. Pipiras is with the Statistics and Operations Research Department, UNC-Chapel Hill, NC, 27599, pipiras@ .unc.edu distribution function. he shift to LRD/H initially occurred in wireline systems 2], 13], and then was followed by similar findings in wireless systems 7], 6], 1], 15]. In particular, there is evidence of LRD/H in wireless internet traffic due to large file sizes from WWW page downloads and multimedia applications including streaming video (music videos, news clips, movies, concerts) and interactive video (e-commerce and gaming). his traffic has bursty behavior affecting network requirements (e.g. capacity and queuesize). hese effects have a significant influence on the network design and are being incorporated in planning for the next generation 4G wireless networks (e.g. 12]). Following this paradigm shift, there has been some recent work on heavy traffic modeling of queueing systems under LRD/H assumptions on the data traffic in wireline systems (see 4], 8], 5]). In these papers reflected fractional Brownian motion (RBM) models for the queue process are obtained, under new state space scalings in the heavy traffic method. he arrival and departure process models in these papers are such that the same scaling of time and state space are natural for both the arrival and departure process; in other words, the same scalings are used in the functional central limit theorems (FCL) and lead to nontrivial (nonzero) limit processes. In this paper, to the best of our knowledge, we give a first result for extending the heavy traffic method under LRD/H assumptions to a wireless queueing system. A one-dimensional queue with wireline arrivals and departures through a wireless medium is considered. We restrict our analysis to a one-dimensional queuing model in order to focus on the development of the heavy traffic scalings, avoiding the complications of the reflected process in higher dimensions. In the 1-D case, the reflection direction is always in the positive queue direction. In higher dimensions, there are complications surrounding the reflected process (unlike the wireline case) due to the possibility of multiple sets of reflection directions on the boundary of the queue state space, each set corresponding to a channel state (see 3] for some furthur discussion). Nonetheless, the scaling results here for the 1-D case are expected to carry over to higher dimensions. More broadly, obtaining the queue limit models, while important in its own right for studying queue characteristics, is an important first step for considering the resource allocation problem. A maor difference from the wireline case is that in the wireless models we consider there are different natural scalings for the FCLs for the arrival and departure processes. In particular, the arrival processes models are from 14] where an infinite source Poisson model is used

2 in a wireline setting. For the infinite source Poisson model, sources turn on according to a Poisson process with intensity (rate) λ, transmit at rate 1, and are on for a random interval which is heavy-tailed with parameter α (more detail is in Section II). he state space scaling for the arrival depends on the growth condition of λ = λ( ) with the scaling parameter : under a slow growth condition on λ( ) (see (15) and (21)) the FCL yields a stable Lévy motion model and under a fast growth condition (see (16) and (2)) the FCL yields a fraction Brownian motion model. However, for the departures process, the state space is scaled according to the rate of channel variations given by parameter (see (1)) and is such that the FCL yields a departure process which converges to a Brownian motion model. For the wireless model, determining the scaling to use in the heavy traffic method for obtaining a queue model is more complex than the wireline case. More precisely, given the operating conditions or characteristics of the arrivals, λ( ) and α, and those of the departures,, the question is: what is the right scaling to use, i.e., according to the arrivals or departures? his is tantamount to the question: given the operating conditions on the arrival and departure, what is the appropriate limit model characterizing the queue content size: reflected Brownian motion (RBM), reflected fractional Brownian motion (RFBM), or reflected stable Lévy motion (RSLM)? his question is answered in this paper; more generally, we give regions of operating conditions where the scalings and resulting limit models apply. What is surprising is that simple conditions emerge for defining these regions; furthermore, a simple inequality (see (23)) relating the channel rate (i.e., ) and heavy-tails in the arrivals (i.e., α) is used in defining some of the regions. his is described in Section II. In Section III is a short summary. In the remainder of this section we briefly highlight the heavy traffic method and results of 3]. hen in Section II we replace the arrival process with an infinite source Poisson model for multimedia traffic and analyze the various queue limit models. Heavy traffic queue model: SRD/L data assumptions. he following is a summary of the heavy traffic wireless queueing model adapted from 3] for a 1-D queue. In particular, an equivalent model is given where time is accelerated using the scaling parameter instead of the rates being scaled where n denotes the scaling parameter (as in 3]). In the heavy traffic method, one considers an embedded sequence of (scaled) queueing systems (Eq. (8) below) indexed by scaling parameter. he mean arrival rate and departure rate of the data in the scaled systems are O( ). he channel state is given by the process L( t) < < 1, (1) so that the rate of channel variations is slower than the rate of arrivals. his is reasonable as the channel coherence times (how long the channel is in a given state) are typically much longer than the service times for packets. he state space scaling is according to the channel process and is given by 1/ γ, where γ = 1 /2. his heavy traffic scaling leads to a RBM limit model which will be seen below. he limit model is obtained under a heavy traffic condition for the wireless model given by two conditions. First, the balance between mean arrival and departure rates Λ a. = λ a v b = J λ d () p()π(), (2) where π is the stationary distribution of the channel, λa is the canonical mean batch arrival rate, v b is the mean batch size, and p() is a nominal power in channel state. Second, any spare capacity beyond that needed to balance the mean arrival process asymptotically goes to zero. In our case, this is given by the reserve power in (14). Applying the heavy traffic scaling with parameter to the queue balance equations, we obtain x (t) = x () + A (t) D (t), (3) where x is the (scaled) size of the queue, A is the cumulative arrival process, and D is the cumulative departure process. he arrival model is given by the following. he l-th interarrival time for the -th system is given by a, l. Let S a, (t) be 1/ times the number of batches that have arrived by real time t, and vl b be the l-th batch size. hen, where M a, (t) = 1 γ A (t) = M a, (t) + S a, (t)] vb S a, (t)] a, a γ l, (4) v b l v b] + vb γ S a, (t)] 1 a, l a is the scaled variations of the arrival about the mean process. Assume that the centered and scaled sequence of processes ] t] w a, 1 (t) = 1 a, l and a w b, (t) = t] 1 v b l v b] (5) for the interarrival times and batch sizes are tight. his is typically the case in the traditional wireline setting under SRD/L assumptions and is appropriate here since the data is arriving to the queues via wireline. Note that the state space scaling γ > 1/2 used here is stronger than the standard 1/ scaling for the wireline case. his implies that M a,n (t). he sum term in equation (4) is the time t in the scaled system plus an error term e (t) that goes to as. he departure process is given by D (t) = 1 γ t I L (s)=} λ d () J p() + u(, ] x (s)) I /2 x (s)>} ds, (6) ]

3 where I } is the indicator function and I x (s)>} constrains the queues from being negative. Equation (6) can be rewritten as D (t) = M d, (t) /2 Λ a t z (t) + 1 t γ I L (s)=} λ d () u(, x (s)) /2 where M d, (t) = 1 (1+)/2 1+ t J J ds, I L(s)=} π() λd () p() ds,(7) and z (t) is the reflection process which represents the work that could have been done using the nominal power p( ), had the queues not been empty. Combining the expressions above and neglecting the error e (t), the prelimit equation is then given by x (t) = x () + M a, (t) M d, (t) + z (t) 1 1+ t 1+ I L(s)=} λd ()u(, x (s)) ds. (8) J he main result in 3] is that (8) weakly converges (in D, ) under the Skorokhod J 1 ]-topology) to the following RBM x(t) = x() + t b(u, x(s)) ds + w(t) + z(t), (9) where w( ) is Brownian motion, z( ) is the reflection process and the drift is given by b(u, x) = λ d ()u(, x)π(). II. SCALINGS FOR LIMI MODELS For the model in Section I the state space scaling 1/ 1 /2, (, 1), was chosen such that the centered departure process converged to Brownian motion. Under the assumptions (5) on the arrival process, this scaling dominated the centered arrival process in the sense that it converged to the zero process. If one chose the state space scaling to be 1/, so that the arrival process converged to a Brownian motion, the resulting limit model would be meaningless since the departure process would diverge. In this section we replace the arrival process by an infinite source Poisson process modeling multimedia traffic. From 14] it is known that the arrival process can be scaled and shown to converge in some (weak) sense to fractional Brownian motion (FBM) or stable Lévy motion (SLM) under a fast growth or slow growth condition, repectively (see (16) and (15) along with (2) and (21) below). he departure process model and scaling will remain unchanged. We will show that there are several possible scalings for meaningful (nondivergent) queue limit models and show which scaling to use depends on the characteristics of the arrival and channel processes. Before discussing the possible queue limit processes, we consider the infinite source Poisson arrival process and queue pre-limit models. Following 14], the fast growth and slow growth conditions are stated along with the state space scaling to be used in each case. A. Background Heavy traffic queue model: LRD/H assumptions. he infinite source Poisson process modeling the arrivals is given by N (s) = 1 Γk s<γ k +V k }, k= where Γ k } k= are Poisson arrivals with intensity λ( ) and X k } k= are i.i.d. positive variables with heavy tail distribution F V (v) = P (V > v) = v α L(v), (1) 1 < α < 2, and L(v) is a slowly varying function at v =. he cumulative arrival process is given by A (t) = 1 λ( ) t N (s) ds. (11) he normalizing factor λ( ) in the above ensures that the mean arrival rate is O( ): EA (t) = t µ on, µ on = EX. he channel process is as before except indexed by : L (s) = L( s), < < 1. Note that the arrival rate is faster than the channel rate, as in the model in Section I. he heavy traffic condition is analogous to that in Section I: a balance between the mean arrival and departure rates µ on = λ()p()π(), (12) and a small reserve power u (x) (see (14)) is assumed (for stability of the queue process). Denoting the state space scaling as β( ), analogous to equation (8) in Section I, we have for the system equation: x (t) = x () + β( ) ( A ) (t) tµ on β( ) β( ) +β( ) 1+ t t t ( 1L(s)=} π() ) p() ds 1 L (s)=} λ()u (x) ds 1 L (s)=} λ()p()1 x (s)=} ds = x () + Ma (t) Md (t) t β( ) 1 L (s)=} λ()u (x(s)) ds + z (t).(13) For fixed, the level of the reserve power is scaled such that its contribution is a drift term in the limit model (see

4 (24)). In particular, u (x) = ū β( ) x > x (14) where ū is a constant. Equation (14) essentially says, for our 1-D queue example, apply all the reserve power u (x) as long as the queue is not empty. Fast and slow growth conditions on arrivals. Some of the possible limit models given below use state space scalings β( ) determined from the arrival process. In particular, the results of 14] are used, where the infinite source Poisson model of the arrival process is shown to converge to SLM or FBM, depending on whether the source rate λ( ) for the arrival process is in the slow growth or fast growth regime. In particular, the regimes are given by slow growth: fast growth: b(λ( ) ) lim lim b(λ( ) ) where b( ) is given by ( ) b(v) = (v), v >, 1 FV = (15) =, (16) and the inverse is g (y) = infv : g(v) y} for a nondecreasing function g. Informally, letting L(v) = 1 in (1) we have F V (v) = v α, b(v) = v 1/α. he slow growth condition is then given by (consistent with (15) for large ) (λ( ) ) 1 α, (17) As in the model in Section I, for the departures we note that 1 ( ) 1 1/2 L(s)=} π() ds (22) converges to Brownian motion in the Skorokhod topology. B. Convergence theorems Here we investigate the possible limit processes for the queue under fast growth and slow growth conditions. In each case the state space scalings lead to either the centered arrival or centered departure being a dominating process where the other process is shown to converge to the zero process. he scaling used and the limit results depend on the parameters of the arrival process (λ( ), α) and departure process () (supposing L(x) = 1 for simplicity). Interestingly, the inequalities α (23) are important for determining the scaling and limit processes in both the fast and slow growth regimes. In the fast growth regime, the limit process is RBM except for a case when the intensity λ( ) lies in a region defined using (23) resulting in a RFBM limit model for the queue (heorem 3 below). In the slow growth regime the limit process is RSLM, except for a case where (23) is used to define a region for λ( ) where the limit process is RBM (heorem 6 below). he inequality is illustrated in Figure 1. he region above line l produces the case for convergence to RFBM in the fast growth case and the region below the line produces the case for convergence to RBM in the slow growth case. or λ( ) α 1 (18) and the fast growth condition by (consistent with (16) for large ) α 1 λ( ). (19) Convergence results for the arrival and departure processes. From 14], the convergence results (used in Section II-B) for the arrival processes are: 1) Under the fast growth condition, λ( ) ( A ( ) µ λ( ) 1/2 (3 α)/2 on ( ) ) (2) L( ) converges to FBM in D, ) with the usual Skorokhod J 1 ]-topology (see 17]). 2) Under the slow growth condition, λ( ) b(λ( ) ) (A ( ) µ on ( )) (21) converges to SLM in finite dimensional distributions and in D, ) with the M 1 -topology (see 17]). Fig. 1. Inequality < α, > α for α (1, 2), (, 1) 2 We suppose for simplicity that the slowly varying function in (1) is L 1 or even L 1 for simplicity. However, the general case is discussed in Remark 1 below. We also note that using standard averaging techniques, the drift term in

5 (14) can be shown to converge (as in 3]) to π() λ()u(x)t. (24) his holds for each case below and we will not comment further on this past heorem 1. Fast growth regime. In this regime, for large, α 1 λ( ) by (16) and the limit models are determined by the inequalities (23) as shown next. heorem 1: Let < α β( ) = = 1. (25) 1+ (1 )/2 hen x converge to RBM. Proof. Part 1: unreflected process. From < α, it follows 2 α+ α 1 λ( ). With β( ) in (25) we have that Md converges to BM this follows from the techniques discussed in the appendix of 3]. here a perturbed test function method using a martingale characterization of Brownian motion was applied (using heorems and in 11]). Noting that for large β( ) λ( )1/2 (3 α)/2 λ( ) we have that M a (t) = 1 α/2+/2 λ( ) 1/2 = 1 α/2+/2 λ( ) 1/2 1, (26) λ( ) ( A ) (t) tµ λ( ) 1/2 (3 α)/2 on (27) converges to the zero process. Indeed, the first term in (27) converges to from (26) and the second term in (27) converges to FBM from (2). he scaling in u (x) is defined such that the corresponding term in (13) (third line down) gives the drift term π() λ()u(x)t in the limit process. Part 2: reflected process. We wish to show z } converges to a reflection process, keeping the queue size nonnegative. A standard approach (see 9] or 1]) is to establish weak convergence to process z (z z) and apply the Skorokhod representation theorem to show z is a reflection process. A key for establishing the weak convergence is showing that the collection of processes z ; } is tight. Following the discussion in 9], section 5.2, an asymptotic continuity argument along with the the reflection map from the Skorokhod problem can be used to show z ; } is tight and we outline the argument for our case next. From (13), define h (t) = x () + Ma (t) Md (t) t β( ) 1 L (s)=} λ()u (x(s)) ds. Due to the convergence of Md to BM, M a to the zero processes, and the fact that x (), λu (x) are bounded, the asymptotic continuity condition holds: } lim δ lim sup P sup sup h (t + s) h (t) µ t Λ s δ =, (28) for each Λ > and µ >. his property implies that z ; } is tight by the following argument. he reflection term z satisfies the z = max }, min s t h (s), (29) where the right hand side is the reflection map (see 9]). From this we have, z (τ + t) z (τ) = max, min x (τ) + h (τ + s) h (τ) ]}.(3) s t he tightness needed for weak convergence of z follows from (3) and the asymptotic continuity in (28) (see 9] for the details of the general criterion for showing tightness). he following two theorems concern the opposite inequality in (23). In this case, the limit processes depend on a finer growth condition on λ( ). heorem 2: Let β( ) = > α 1+ = 1 (1 )/2. Furthermore suppose that λ( ) is such that for consistency α 1 2 α+ λ( ). (31) hen x converges to RBM. Proof. he argument is the same as before where we only need to note that under (31) we still have that the first term in (27) converges to. heorem 3: Let β( ) = > α λ( ). (32) λ( ) 1/2 (3 α)/2 Furthermore, suppose that λ( ) is such that α 1 λ( ) 2 α+. (33) hen x to converges to RFBM.

6 Proof. Using the β( ) in (32) and noting (2), we have that Ma converges to FBM. Note that hen β( ) (1 )/2 = M d (t) = β( ) (1 )/2 1 (1+)/2 1+ t λ( )1/2 1. (34) 1 α/2+/2 ( 1L(s)=} π() ) λ()p() ds(35) converges to the zero process. Indeed, the first term in (35) converges to from (34) and the second term in (35) converges to BM from (22). he asymptotic continuity approach for obtaining the reflected process (described in heorem 1) can be applied here due to the continuity properties of the paths of fractional Brownian motion. Slow growth regime. In this regime, for large, λ( ) α 1. Limit models for subclasses obtained using the inequality illustrated in Figure 1 are given in the following theorems. For heorems 4 and 5 we note that the arrivals dominate departures and queue process converges to RSLM. heorem 4: Let β( ) = > α hen x converges to RSLM. λ( ). (36) λ( ) 1/α 1/α Proof. From > α it follows that (α 1)2 < 1 α 2 + α 2 and we have λ( ) α 1 1 α/2+α/2 α 1. With the β( ) in (36), we have from (21) that Ma converges to SLM. We note that β( ) (1 )/2 = = λ( ) 1 1/α 1/α+/2 1/2 ( λ( ) 1 α/2+α/2 α 1 ) α 1 α 1. (37) From this it follows, as in the discussion of heorem 3, that Md converges to the zero process. A remaining issue left to work out are the details for convergence of the reflection process since SLM is a discontinuous process; consequently, the asymptotic continuity approach described in the proof of heorem 1 cannot be applied. he following two theorems concern the opposite inequality in (23). As in the fast growth regime, the limit processes depend on a finer growth condition on λ( ). heorem 5: Let < α β( ) = λ( ). (38) λ( ) 1/α 1/α Furthermore, suppose that λ( ) is such that λ( ) 1 α/2+α/2 α 1 << α 1. hen x converges to RSLM. Proof. In this case we have that (37) still holds and so the discussion in heorem 4 carries over to this case. heorem 6: Let < α 1 β( ) =. (39) (1 )/2 Furthermore, suppose that λ( ) is such that λ( ) 1 α/2+α/2 α 1 α 1. hen x to converges to RBM. Proof. Under the β( ) in (39) and using (22), M d converges to Brownian motion. Note that β( ) λ( )1/α 1/α λ( ) so that by (21) we have that Ma process. = 1/α+/2 1/2 λ( ) 1 1/α 1, (4) converges to the zero Remark 1: We assumed for simplicity in heorems 1-6 that the slowly varying function appearing in (1) is L 1. However, the conditions determining the various limit processes can also be formulated for arbitrarily slowly varying functions L(v) (the most general case). For example, as shown in 14], fast growth is equivalent to α 1 λ( ) L( ). hen one can show that heorems 1-3 continue to hold if (31) is replaced by α 1 2 α+ L( ) 2 λ( ) L( ) and (33) is replaced by Remark 2: If α 1 λ( ) L( ) 2 α+ L( ) = α, then, loosely speaking, the limit process is expected to be a reflection of the combinations of Brownian motion, stable Lévy motion, or fractional Brownian motion.

7 Fig. 2. Summary of possible limit models. III. SUMMARY Figure 2 summarizes heorems 1-6. Note that the inequalities α relating the channel and arrival process are used in the fast growth and slow growth regimes. hey define regions for the growth conditions on λ( ) which cover the various possible limit models for the queue dynamics. o the best of our knowledge, these are the first results of their kind for wireless systems and have no analog in the current work in wireline models. Obtaining the queue models is an important step in characterizing the queue behavior and then considering the resource allocation problem in networks with competing queues. Our future work will broadly be in this direction, with a next focus on the details of RSLM and characterizing the queue behaviors using the collection of limit models obtained here. REFERENCES Personal Multimedia Communications (IEEE), Proceedings, 2(22), pp ] J. BERAN, R. SHERMAN, M.S. AQQU, AND W. WILLINGER, Longrange dependence in variable-bit-rate video traffic, IEEE ransactions on Communication, 43(1995), pp ] R.. BUCHE AND H.J. KUSHNER, Control of mobile communications with time-varying channels in heavy traffic, IEEE rans. Automatic Control, 47 (22), pp ] K. DȨBICKI, M. MANDJES, raffic with an fbm Limit: Convergence of the Stationary Workload Process, Queueing Systems, 46 (24), pp ] R. DELGADO A reflected fbm limit for fluid models with On/Off sources under heavy traffic, Stochastic processes and their applications (26), (in press) doi:1.116/.spa ] M. JIANG, M. NIKOLIC, S. HARDY, L. RAJKOVIC, Impact of selfsimilarity on wireless data network performance, IEEE Intl. Conference on Comm., 2, pp , 21. 7] R. KALDEN, S. IBRAHIM Searching for self-similarity in GPRS, Proceedings of the 5th annual Passive and Active Measurement Workshop (PAM 24), Antibes Juan-les-Pins, France, April 19-2, 24. 8]. KONSANOPOULOS, S. LIN Fractional Brownian Approximations of Queueing Networks, Stochastic Networks, Lecture Notes in Statistics, 117, Springer, NY, ] H.J. KUSHNER, Heavy raffic Analysis of Controlled Queueing and Communication Networks. Springer, New York, 21. 1] H.J. KUSHNER, P. DUPUIS, Numerical Methods for Stochastic Control Problems in Continuous ime, Second Edition, Springer, New York, ] H.J. KUSHNER Stochastic Approximation Algorithms and Applications. Springer, NY ] A. KRENDZEL, Y. KOUCHERYAVY, J. HARJU, S. LOPAIN, Network Planning Problems in 3G/4G Wireless Systems, he 1st COS 29 Management Committee Meeting, Malta, Oct ] W.E. LELAND, M.S. AQQU, W. WILLINGER, AND D.V. WILSON, On the self-similar nature of Ethernet traffic, IEEE/ACM ransactions on Networking, 2:1 (1994), pp ]. MIKOSCH, S. RESNICK, H. ROOZÉN, Is Network raffic Approximated by Stable Lévy Motion or Fractional Brownian Motion? he Annals of Applied Probability, 17, No. 1, pp ] R. NARASIMHA AND R. RAO, Modeling Variable Bit Rate Video On Wired and Wireless Networks Using Discrete-ime Self-Similar Systems, Proceedings, IEEE International Conference on Personal Wireless Communications, 22, pp ] A. L. SOLYAR, Max Weight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic, he Annals of Applied Probability, 14(1): 1 53 February ] W. WHI, Stochastic Process Limits Springer, New York, 22. 1] D.R. BASGEE, J. IRVINE, A. MUNRO, P. DUGENIE, D. KALESHI, O. LAZARO, Impact of Mobility on Aggregate raffic in Mobile Multimedia System, he 5th International Symposium on Wireless