KLEINROCK, L. 1975. Queuing Systems. Wiley, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....other, in the absence of interference from competing transmissions. with the transmitters using the maximum available power, are connected by straight lines. Nodes that can communicate directly but are not very close are more sensitive to competing transmissions. B. When not to FIFO Following [14], we refer to the set of rules that governs the order with which a node attempts to forward the packets in its queue as the node s queuing discipline. The most common queuing discipline is the First In First Out (FIFO) rule, i.e. the node attempts to forward first the packet that was first in the ....

L. Kleinrock, Queuing Systems, Volume I: Theory, New York: John Wiley and Sons, 1st edition, Jan. 1975.

....of packet drops, maximum delay, delay jitter or bandwidth. Much research was done in the early days of computer networking comparing circuit switching, packet switching and message switching (a variant of packet switching, in which the whole information flow is treated as a single switching unit) [96, 10, 164, 97, 175, 95]. Most of the work was done in the context of packet radio, satellite, and local area networks and shows how in these environments packet switching provided higher throughput for a given bound on the average delay. Packet switching not only made an effective use of the network bandwidth, but it ....

....3.3, but for the core of the network. These analytical results do not include many network effects that may affect the response time, and so Section 3.5 uses ns 2 simulations to validate the results for the core. Section 3.7 concludes this chapter. Early work in the late 60s and in the 70s [96, 10, 164, 97, 175, 95] studied the response time of packet and circuit switching in the context of radio networks, satellite communications and the ARPANET (the precursor of the modern Internet) These three network scenarios had something in common: links had less capacity than the bandwidth that an end host could ....

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Leonard Kleinrock. Queuing Systems, Volume I: Theory, volume 1. WileyInterscience, New York, 1975.

....Not only are we faced with the queuing problems that arise from the stochastic nature of the demands; we are also faced with the issue of allocating resources to a geographically distributed (and possibly mobile) set of demands. Were we not in this distributed environment, queuing theory [6] would provide us with the ultimate mean response time throughput performance profile. However, we have additional loss of resources due to the cost of organizing the separated demands into some kind of cooperating queue, which permits intelligent access to the available resources. For the ....

....that power is maximized at that point on the response time throughput profile where a straight line from the origin first becomes tangent to the profile; in Fig. 2 we denote the optimum throughput operating point by g . What is amazing about this result is that for all M G 1 queuing systems [6], this point occurs where E[N] the average number of messages in the system, is exactly one What makes this interesting is that it is intuitively the correct operating point for deterministic systems [11] GIANT STEPPING In the case of packet radio systems, a totally different consideration ....

L. Kleinrock, Queuing Systems, Volume I: Theory, Wiley, New York, 1975.

....has a simple analytical model shown in Figure 4.4. The steady state probabilities Pi are easily determined using standard methods and the reservation rejection probability Pn k, is just the steady state probability of state n k. Such an analysis yields the following equations using results from [44]. Pm= P0 H . min i 1, n i=0 t (4.1) n t n Lt iO n Lt (i DO Figure 4.3: Analytical Model for Exponentially Distributed Defer Bounds Figure 4.4: Analytical Model Approximating Constant Defer Times 96 Thus we get: where: m m n m n (4.2) Xtg)m Y] n (Xtg)m (4.3) P0= l y] m . m ....

Kleinrock L. Queuing systems, Volume 1: Theory. John Wiley 2 Sons, April 1975.

....queuing model for filtering with flow caching. where t is the response time or the total time of a packet traversing the router. PT(t r) is the tail distribution of t when t exceeds r. r and 8 are design parameters. The response time of the tail distribution for M M 1 queue can be written as [4]: PT(t r) exp( g(1 ix h)r) 2) A reasonable assumption is to choose 8 so that when g0 = 1 r (i.e. a packet is processed with little queuing delay) the equality in Eq. 1) holds. For instance, if we set h = 1 Mpps and x = 3.3 Mpps, we find that 8 = 1.0. Now, let us model flow caching ....

L. Kleinrock, Queuing Systems, vol. I, Wiley, New York, 1975.

....the downlink FIFO queue at the base station by embedding the system at the end of the packet block transmission period, T E . This interval is available to transport a packet block with probability 1 r (behavior frame to frame independent) Let T(z) denote the Probability Generating Function (PGF) [7] of the transmission time (geometrically distributed) of a packet block: 1 rz z r = 5) The following derivations made for a single resource could be extended to the case of n resources (with sufficient accuracy for medium heavy traffics) by considering that the information content ....

....(with sufficient accuracy for medium heavy traffics) by considering that the information content sent in T E is multiplied by n. A. Poisson Traffic The mean message delay is obtained by adopting an M G 1 model for the transmission queue. From the PollaczekKhintchin formula and the Little theorem [7], we have: s 1 1 2 1 # # # # # # FIFO FIFO w E A A M T t E (6) where A FIFO (1) and A FIFO (1) are respectively the first and second derivative (computed for z = 1) of A FIFO (z) the PGF of the number of message arrivals in a multiframe: 1 1 ....

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L. Kleinrock. Queuing Systems (Vol. I and II), J. Wiley & Sons, N.Y., 1976.

....(y) for j = 1 (case a) and j = 2 (case b) with the following crisp input values: PLR V i = 6.310 6 and typePi = 0. D. Management of Web Tra#c Sources The remaining slots within V (if any) after having assigned video tra#cs plus W slots are cyclically assigned to Web sources. According to [17], a round robin scheme (actually, its ideal version called processor sharing) is more convenient than the First Input First Output (FIFO) discipline to manage the transmission of Poisson arrivals when the distribution of their transmission times, T , permits to have E[T ] 2E[T ] This ....

L. Kleinrock. Queuing Systems. J. Wiley & Sons, N.Y.,1976.

....a Poisson distribution because the probability that an LFE queue is full depends on the LFE load, but, for the sake of simplicity, we approximate it with a Poisson distribution as follows: the LFE queue is an M=M=1=n queue. Thus, the probability of a packet arriving at a full LFE queue is (see [11]) b ( LFE ) Pn = LFE= LFE : 1) Thus, we assume the fraction of trac o = b LFE = r b i to be sent to the MFE pool due to LFE overload. Furthermore, as in [7] even if a packet is being processed by an LFE, with a xed probability b the LFE will not be able to nd the ....

....the sum of nonprocessed packets from the LFEs, together with the pre scheduled packets. As the part of workload sent for resolution to the MFE represents a sum of Poisson processes, the sum is a Poisson process as well. This workload and the corresponding M=M=m queue waiting time are on average [11]: sum = k ( p o rf ) 5) T 2 = MFE PQ sum (1 ) 6) where = sum m MFE ; 7) P 0 = m 1 P j=0 j j ; PQ = P 0 (m ) 8) 2.3 Switch Model General input output switch. The switch is characterized by two parameters the switch port speed s and the number ....

L. Kleinrock. 1975. Queuing Systems. John Whiley & Sons.

....a Poisson distribution because the probability that an LFE queue is full depends on the LFE load, but, for the sake of simplicity, we approximate it with a Poisson distribution as follows: the LFE queue is an M=M=1=n queue. Thus, the probability of a packet arriving at a full LFE queue is (see [11]) b ( LFE ) Pn = LFE= LFE : 1) Thus, we assume the fraction of trac o = b LFE = r b i to be sent to the MFE pool due to LFE overload. Furthermore, as in [7] even if a packet is being processed by an LFE, with a xed probability b the LFE will not be able to nd the ....

....the sum of nonprocessed packets from the LFEs, together with the pre scheduled packets. As the part of workload sent for resolution to the MFE represents a sum of Poisson processes, the sum is a Poisson process as well. This workload and the corresponding M=M=m queue waiting time are on average [11]: sum = k ( p o rf ) 6) T 2 = MFE PQ sum (1 ) 7) where = sum m MFE ; 8) P 0 = m 1 P j=0 (m ) j j (m ) m ; PQ = P 0 (m ) m : 9) 2.3 Switch Model General input output switch. The switch is characterized by two parameters the switch port speed s ....

L. Kleinrock. 1975. Queuing Systems. John Whiley & Sons.

....characterized, in addition to the offered load, by the distributions of the message interarrival times and message lengths. In our experiments we considered exponential (coefficient of variation C equal to one) hypoexponential (C 1) and hyperexponential (C 1) interarrival time distributions [KLEI75]. Furthermore, IThe confidence level was chosen to be equal to 90 . 19 the message length Msg was assumed to be constant in each experiment and equal to 1, 4, and 8 cells. In undefioad conditions the performance measure of interest in our analysis is the station average access delay, i.e. the ....

L. Kleinrock, "Queuing Systems", Volume I: Theory, John Wiley & Sons, 1975.

....The third random variable in the message response time, as discussed in Section 1, is the resequencing (R ) delay. The message response time is then MRT = W T R: 2) 9 nding n messages in the subsystem. The state occupancy probabilities for the M M N queue at steady state are given by [15]: n N N N n N n N (5) where p 0 = N 1 # 1 : 6) The expected message response time is given by: E[MRT ] E[W ] E[T ] E[R] 7) Independent of the communication paradigm, E[W ] is given from M M N analysis [1] E[W ] ....

....other hand, does depend on the communication paradigm. Analysis of the resequencing problem for FIFO channel communication has been considered in the literature by several authors. Yum and Ngai [19] studied the resequencing of messages 10 ned subsystem will achieve steady state for = N 1 [15]. Let p n be the steady state probability of ne , the system utilization, as: N = N when = 1:0: 4) The de nite number of messages to be transmitted. The system up to, but not including, the resequencing bu er is an M M N queue. We de Once again, consider the system shown in ....

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Kleinrock, L. Queuing Systems, vol. 1. John Wiley & Sons, Inc., 1975.

....capacity at every network level (i.e. C I , CR , CN , and C) We assume that the most congested network link from level l to the clients is the link at level l. Delays on network levels lower than l are neglected. Let S be the average document size of all N documents. The M D 1 queuing theory [17] gives for the delay at every network level in a caching hierarchy, and for the delay at every network level in a distributed caching scheme. IV. HIERARCHICAL VS DISTRIBUTED CACHING: NUMERICAL COMPARISON To quantitatively compare hierarchical caching against distributed ....

L. Kleinrock, Queuing Systems, Volume I: Theory, Wiley, 1975.

....Multiple access techniques for communication have been interesting topics for a variety of investigations and implementation challenges. In recent years, random access packet switching has been developed as a useful multiple access technique for systems with high peak to average tra#c ratios [8]. In a slotted system, the channel time is divided into slots of duration equal to the transmission time of a packet, and each user is allowed to start transmission of a packet only at the beginning of a slot [4,10,18,17] More recently, we have seen the introduction of spread spectrum schemes as ....

L. Kleinrock, Queuing Systems, Vol. I: Theory, WileyInterscience, New York, NY, 1975.

....traffic. Assume that audio packets share alone the buffer K. This can be the case of a bottleneck crossed only by audio packets, or the case of a bottleneck router implementing a per flow or a per class queueing. Thus, for ae 1, the loss probability of an audio packet in steady state is given by [15]: ae) 1 Gamma ae 1 Gamma ae K 1 ae K ; 1) and for ae = 1 it is equal to (ae) 1 K 1 : Now, we add redundancy to each packet in a way that if a packet is lost, it can be still partially retrieved if the packet containing its redundancy is not lost. The redundancy is located ....

L. Kleinrock, Queuing systems, John Wiley, New York, 1976.

....other coscheduling algorithms. First, the simulation environment and the metrics used are presented. Then, the results obtained in the simulations are described and commented on. 5.1 Simulation and Metrics Every node of the cluster has been simulated as shown in fig. 3. In this model, based on [26], when a tasks quantum is expired, the task is removed from the CPU and is reinserted in the RQ whenever the task has not finished all its requesting time. If a task does not expire its quantum due to a communication primitive or a page fault requesting service, it will be inserted in the SLEEP ....

L. Kleinrock. "Queuing Systems". John Wiley and Sons, 1976.

....We will only attempt to brie y review the most relevant results concerning multiclass 1 queuing networks. We will also brie y review some relevant results from the eld of packet routing. For basic de nitions in queuing theory we refer the reader to standard texts such as Kelly [29] Kleinrock [28] and Walrand [52] The reader may wish to skip this section and proceed directly to the new de nitons in Section 3. 2.1 Multiclass queuing networks We can view dynamic packet routing as one restrictive (but still quite important and non trivial) type of multiclass queuing network. In particular, ....

L. Kleinrock. Queuing systems. Wiley, New York, 1975. 23

....and let L ARQ (s) L BRQ (s) and L DRQ (s) denote the Laplace transforms of the pdfs respectively. We can present the following equations: d RQ (t) r b RQ (t) 1 r ) a(t) b RQ (t) 3 5) and L DRQ (s) r L BRQ (s) 1 r ) L A (s) L BRQ (s) 3 6) 9] l rq here, L A (s) 3 7) 9] [10] l rq s l rq = l. Substituting (3 4) and (3 7) for L A (s) and L BRQ (s) in equation (3 6) ln 1 (1 P t (l) l rq ln 1 (1 P t (l) L DRQ (s) r (1 r) ln 1 (1 P t (l) s l rq s ln 1 (1 P t (l) s l rq = l rq s 8 = L A (s) 3 8) Equation (3 8) shows that the interdeparture ....

....(n 1) the so called interleaving feedback for further study. Based on the delay model of the protocol, we can treat the CR component (RQ) as an M M 1 system. Apply the average service rate rq derived in equation (3 4. 1) and the input rate l to the M M 1 queue, we achieve the average delay of RQ [10], 1 E[t rq ] 3 13) ln 1 ( 1 P t (l) l We have proved that the input traffic process to TQ, or say the output traffic from RQ, is a Poisson process. We also assume the size of a slot is one. For TQ, the service time for a packet is constant, one slot. So, following M D 1 queue ....

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Leonard Kleinrock, "Queuing Systems", John Wiley and Sons INC., NY., 1976

....Assume that audio packets share alone the buffer . This can be the case of a bottleneck crossed only by audio packets, or the case of a bottleneck router implementing a per flow or a per class queueing. Thus, for , the loss probability of an audio packet in steady state is given by [15]: # (1) and for ( it is equal to ) Now, we add redundancy to each packet in a way that if a packet is lost, it can be still partially retrieved if the packet containing its redundancy is not lost. The redundancy is located packets ....

L. Kleinrock, Queuing systems, John Wiley, New York, 1976.

....for additional bandwidth divided by the rate at which they are met for a single client, and m is the number of units of 2. 5 Mbps bandwidth and p k is the probability of k requests in the queue, then the steady state probabilities of queue occupancy for a multiserver, nite population queue are from [15]: P k # p 0 # k 2N k # # # 0 # k # m# 1 p 0 # k 2N k # # k# m# m m#k # m # k # 2N # # # # # # # # # #2# and p 0 # # m#1 k#1 2N k # # # k # # m#1 m#1 # k k# m# m m#k # ##1 # #3# If 1 # is the mean time between requests for bandwidth and 1## the ....

Kleinrock, L. (1975) Queuing Systems, Vol 1, John Wiley and Sons Inc.

....i s are di erent. Obviously, fr 1 ; r k g R. De nition 1 The branch system P i is an M jDj1 queue of Poisson input rate and xed service rate r i . Let N i be its expected system population, in the steady state. 2 Remark: By the well known Pollaczek Khinchin formula for M jDj1 (see [6], vol. I, p. 188) N i = r i 1 r i 2 r i 2 2 1 r i De nition 2 Let b i 1 (i = 0; k 1) be the steady state probability that the number of packets j in the switch (bu er and in service) is such that q i j q i 1 . 2 Note. b i is the probability that ....

....(sketch) Each branch system contributes to the other branches in p( while this is not allowed in the considered switch S. But then the system S contributes a greater variance (than S) to the steady state. The Theorem then follows from the Pollaczek Khinchin mean value formula (see e.g. [6], Vol. I, p. 187) for any M jGj1 system (for general service time distribution this formula gives N = x ( x) 2 1 2 b x 2 2(1 x) where x is the expected service time of the switch and 2 b its variance) 2 Note. A similar phenomenon is happening when we compare the ....

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L. Kleinrock, \Queuing Systems", Vol I, Wiley eds.

....level. Most of the tools do not address the end to end performance analysis that takes into account the effect of application, protocols, and network on the system performance. Queueing network analytic models concerning standard data and communication networks can be found in numerous references [8 11]. In this paper we present a generic three level hierarchical approach for analyzing the end to end performance of an application running on an HPCC system. The overall system is partitioned into application level, protocol level and network level. Functions at each level are modeled using ....

L. Kleinrock, Queuing Systems, Vol. 2: ComputerApplications, Wiley, New York 1976.

....but the average of four 1 second requests and one 46 second request is 10 seconds; the average response time grows quickly. Little s Law states that the number of requests in service, N, equals the product of the request arrival rate, l, and the total time, W, that a request remains in service [Kleinrock]. N=lW At 15 URLs per second, Little s Law says that that for every Mbit s of WAN traffic, there are 150 concurrent client connections. Hence, for every Mbit s of WAN traffic, we need 5 MB for TCP buffers, 10 MB for client data structures, and 10 40 MB for NetCache data structures. Deploy 50 MB ....

Leonard Kleinrock. Queuing Systems, Volume 1, Wiley, 1975.

....state to state having constant transition rates, i.e. 7) Note that the origin is now state and is in . Furthermore, for an M M 1 queue, the optimal loss likelihood is obtained by substituting the optimal slope from Section 3.4 in Eq. 7) 8) Recall the steady state probability , see e.g. [15]. Then, a very interesting observation is made: is equal to the probability ratio in a constant rate case Is there a similar relation between and when the arrival and departure rates are no longer constant Proposition 1: Let and . Then the probability of an optimal path, from the ....

L. Kleinrock. Queuing Systems, Theory, volume I. John Wiley, 1975.

....models of FMS s, and find pooling to generally increase system throughput. Calabrese [14] extends this result to open queuing network models of job shops and shows that work in process inventories can be reduced by 5increasing machine pooling. Smith and Whitt [43] Benjaafar [4] Kleinrock [33] and Gross and Harris [22] offer a more general discussion of the effect of resource pooling in queuing systems. Buzacott [6] examines the effect of machine and routing flexibility using a stochastic model with two part types and find setup times and product mix variety to limit the effectiveness ....

Kleinrock, L., Queuing Systems, Vol. I, John Wiley, New York, New York, 1975.

....measured for eight of the Perfect Club R fl codes. The measurements were performed using various assumptions about instruction latencies: The left three columns assume unit instruction latencies, and the right three columns 4 The astute reader will recognize this as a version of Little s Law [19] from queuing theory. 15 Table 2: Average Instruction Latency Measurements INT=1 FLOAT=1 INT=1 FLOAT=6 load latency load latency benchmark 1 10 100 3 30 300 ADM 1.000 2.870 21.566 3.124 8.733 64.823 QCD2 1.000 2.756 20.316 3.111 8.379 61.060 MDG 1.000 3.142 24.560 4.237 10.662 74.915 ....

Leonard Kleinrock. Queuing Systems, Volume I: Theory. John Wiley and Sons, 1975.

....proposed for the general performance evaluation of manufacturing systems [2] 3] 10] none of these models deals explicitly with the relationships between batch sizes and performance. In the queueing literature, a significant body of work exists on queues with bulk arrivals and bulk service times [8]. However, exact results exist only for simple models. None of these models account for setup times between batches and or the possibility of alternating priority scheduling. In the production and manufacturing literature, Karmarkar [4] was the first to examine, using a queuing model of a single ....

Kleinrock, L., Queuing Systems, Vol. I, John Wiley, New York, New York, 1975.

....capacity at every network level (i.e. C I , CR , CN , and C) We assume that the most congested network link from level l to the clients is the link at level l. Delays on network levels lower than l are neglected. Let S be the average document size of all N documents. The M D 1 queuing theory [17] gives E[T h t jl] S C l Gamma fi h l Delta S Delta (1 Gamma fi h l Delta S 2C l ) for the delay at every network level in a caching hierarchy, and E[T d t jl] S C l Gamma fi d l Delta S Delta (1 Gamma fi d l Delta S 2C l ) for the ....

L. Kleinrock, Queuing Systems, Volume I: Theory, Wiley, 1975.

....at every network level (i.e. C I , CR , CN , and C) We assume that the most congested network link from level l to the clients, is the link at level l. Delays on network levels lower than l are neglected. Let S be the average document size of all N documents. The M D 1 queuing theory [13] gives E[T h t jl] S C l Gamma fi h l Delta S Delta (1 Gamma fi h l Delta S 2C l ) 6 for the delay at every network level in a caching hierarchy, and E[T d t jl] S C l Gamma fi d l Delta S Delta (1 Gamma fi d l Delta S 2C l ) for ....

L. Kleinrock, Queuing Systems, Volume I: Theory, Wiley, 1975.

....queue. Figure 11 shows the average number of packets waiting in the spy queue as a function of the offered load. As the saturation arises (offered load : 0. 3) the number of packets in the queue becomes close to the maximum (the size of the queue) This behavior is typical of the queuing systems [11]. It occurs when the rate (classically noted ae) becomes larger than 1. In both cases (adaptive or Z 2 routing) curves are very similar. It is interesting to observe the behaviors for the waiting time in the spy queue (figure 12) This figure shows the average waiting time for the packets in ....

Kleinrock, L. Queuing systems, vol. 1. John Wiley & Sons, 1976. 17

....will only attempt to briefly review the most relevant results concerning multiclass 1 queuing networks. We will also briefly review some relevant results from the field of packet routing. For basic definitions in queuing theory we refer the reader to standard texts such as Kelly [29] Kleinrock [28] and Walrand [52] The reader may wish to skip this section and proceed directly to the new definitons in Section 3. 2.1 Multiclass queuing networks We can view dynamic packet routing as one restrictive (but still quite important and non trivial) type of multiclass queuing network. In ....

L. Kleinrock. Queuing systems. Wiley, New York, 1975.

....of flows is allowed to change dynamically. The only assumption we make about the service discipline employed by the switch is that the packets of each flow are served in FIFO order. Thus, the switches could be strict FIFO, FIFO , Priority, Stop and Go, Fair Queuing, etc. 4] 13] 33] 28] [29]) Finally, we assume that the link capacity is the bottleneck resource in the network. However, we believe that the results of this work are equally applicable if host or switch processing capacity is the bottleneck. 1.3 Optimality Criterion This work chooses the so called maxmin or bottleneck ....

Kleinrock, L., "Queuing Systems", vol.2., New York, Wiley, 1976.

....The set of flows is allowed to change dynamically. The only assumption we make about the service discipline employed by the switch is that the packets of each flow are served in FIFO order. Thus, the switches could be strict FIFO, FIFO , Priority, Stop and Go, Fair Queuing, etc. 4] 13] 33] [28], 29] Finally, we assume that the link capacity is the bottleneck resource in the network. However, we believe that the results of this work are equally applicable if host or switch processing capacity is the bottleneck. 1.3 Optimality Criterion This work chooses the so called maxmin or ....

Kleinrock, L., "Queuing Systems", vol.1., New York, Wiley, 1975.

....n l # # l # n l # n l j=0 1 j # # l # j . 5) Let the probability that there are i high priority clients and j low priority clients served by the common pool partition be P (i, j) The probability distribution of P (i, j) can be approximated by the technique of reduced Markov chain [9]. In Eq. 6, the first term at the right hand side indicates the probability of having i high priority clients, and the second term indicates the probability of having j low priority clients, given i high priority clients in the common pool partition. P (i, j) 1 i # #h # i # nm k=0 1 ....

Kleinrock L (1975) Queuing Systems, Vol. 1: Theory. John Wiley & Sons, Chichester

....or is exponentially distributed by studying the variance of object updates. Once they have determined if the object is updated periodically or exponentially, they can use the time difference between object requests and the last modified time stamps to estimate the average update period [11]. 9 0 2000 4000 6000 8000 10000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 D (sec) PDF (a) Spanish Newspaper Web Site 0 2000 4000 6000 8000 10000 0 0.05 0.1 0.15 D (sec) b) BBC News Web Site Figure 5: Distribution of object update intervals. 10 day logs. Servers are polled every ....

L. Kleinrock, Queuing Systems, Volume I: Theory, Wiley, 1975.

....in addition to the offered load, by the distributions of the message interarrival times and message lengths. In our experiments we considered exponential (coefficient of variation C equal to one) hypoexponential C ( 1 , and hyperexponential C ( 1 interarrival time distributions [KLEI75]. Furthermore, 1 The confidence level was chosen to be equal to 90 . 20 the message length Msg was assumed to be constant in each experiment and equal to 1, 4, and 8 cells. In underload conditions the performance measure of interest in our analysis is the station average access delay, i.e. the ....

L. Kleinrock, "Queuing Systems", Volume I: Theory, John Wiley & Sons, 1975.

.... l j l l n j j n n l l l l n y probabilit l l l l j (5) Let the probability that there are i high priority clients and j low priority clients served by the common pool partition be P(i, j) The probability distribution of P(i, j) can be approximated by the technique of reduced Markov Chain [9]. In equation 6, the first term at the right hand side indicates the probability of having i high priority clients, and the second term indicates the probability of having j low priority clients, given i high priority clients in the common pool partition. 0 0 1 1 1 1 = ....

Kleinrock, L. (1975) Queuing Systems, Vol. 1: Theory. John Wiley and Sons.

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KLEINROCK, L. 1975. Queuing Systems. Wiley, New York.

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Kleinrock, L., Queuing Systems, Vol. I: Theory, John Wiley, 1975. Capacity Planning for Web Services: metrics, models, and methods, Prentice Hall, Upper Saddle River, NJ, 2002. Scaling for E-business: technologies, models, performance, and capacity planning, Prentice Hall, Upper Saddle River, NJ, 2000.

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L. Kleinrock, Queuing Systems, Volume I: Theory, WileyInterscience, NY, 1975.

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L. Kleinrock. Queuing Systems. John Wiley & Sons, 1975.

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L. Kleinrock (1976), Queuing systems, Volume I. John Wiley, New York. P. Kolesar (1970), `A Markovian model for hospital admission and scheduling ', Management Science, 16, pp. 384-396.

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L. Kleinrock, Queuing systems, John Wiley, New York, 1976.

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Leonard Kleinrock. Queuing Systems, volume 2. John Wiley, New York, 1975.

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L. Kleinrock. Queuing Systems, Volume I: Theory.J.Wiley & Sons, New York, 1995.

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L. Kleinrock, "Queuing Systems", Vol.2: Computer Applications, John Wiley & Sons, 1976

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L. Kleinrock, "Queuing Systems", Vol. 1, Wiley, New York, 1975

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L. Kleinrock, Queuing Systems, vol.I, Theory, John Wiley & Sons, New York, 1976.

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L. Kleinrock. Queuing Systems, Volume 1: Theory. John Wiley and Sons, 1975.

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L. Kleinrock, Queuing Systems. John Wiley & Sons, 1975.

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Kleinrock, L. (1975). Queuing Systems, Vols. I and II, Wiley, New York.

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