- Email: [email protected]
Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches
© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue
Letter Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches Klaus-Dieter Langer Abstract: In this letter, we present an exact closed-form solution for the probability distribution of the packet batch-size at the output ports of unbuffered crossbar based packet switches. The general case of a switch with M inputs and N outputs is considered under balanced traffic and using an optimum arbitration scheme. The solution can be applied to a wide range of queuing systems, even beyond the field of packet switching. Keywords: Unbuffered crossbar, Packet switching, ATM, Discretetime analysis, Batch-size distribution
1. Introduction We consider a unicast packet switch with M inputs and N outputs, where incoming and outgoing links are assumed to run at the same speed. Switching is performed by a nonblocking M × N interconnection network; throughout this paper, we simply assume a crossbar. Furthermore, we consider the special case where there are no buffers for storing packets inside the system, and where from each arriving batch a maximum number of packets will be selected for switching by an appropriate arbitration unit. This is done on basis of random decisions, if the batch contains more than one packet destined for identical outputs. Requests that are not accepted due to output contention (i. e. external blocking) are dropped. Architectures like this are suitable e. g. for interconnecting multiple processors to shared memory modules, or for photonic switching, where buffering of packets in the optical domain (via delay lines) or moving them to electronics for buffering is avoided for the sake of simplicity. Performance measures of this very fundamental system, which is clearly the simplest case of a packet switch, represent the lower bounds for numerous more sophisticated switches such as input-buffered or output-buffered crossbars. The throughput of an unbuffered crossbar switch in case of balanced traffic load and optimum service, derived by Patel in his well-known paper [1], is γ = 1 − (1 − λ/N) M , {M, N} ∈ N+ . For switches with a large number of ports, the throughput converges to lim M,N→∞ γ = 1 − e−λM/N . Apparently, a closed-form solution for the probability distribution of the departing
Received November 25, 2002. Revised Februar 18, 2003. K.-D. Langer, Fraunhofer Institut für Nachrichtentechnik, HeinrichHertz-Institut, Department of Optical Networks, Einsteinufer 37, D-10587 Berlin, Germany. E-mail: [email protected]. de ¨ 57 (2003) No. 5, 351−354 Int. J. Electron. Commun. (AEU)
packet batch-size is not known in literature; an approach for deriving recursively the output port occupancy distribution can be found in [2]. This letter provides an exact and easy to apply closed-form solution based on the Stirling numbers of the second kind. The paper is organised as follows. First, we introduce the traffic model, and then we analyse the batch-size of packets leaving the switch. In addition to the corresponding discussion, a verification of the main result is given in the Appendix.
2. Assumptions and analysis The following assumptions are the same as used in [1]. We consider a discrete-time model with M inputs and N servers (output ports). Time is divided into time-slots of constant length corresponding to the transmission time of the fixed-sized packets to be switched. We suppose synchronous operation, where all requests arrive at time-slot boundaries. Since the service of a packet, i. e. the transfer from the input to the desired output port, takes exactly one time-slot, it follows that a packet, if it was accepted by the switch, has left the system at the time-slot boundary next to the arrival instant. Notice that the analysis is not bound to a specific discrete-time model such as the Early Arrival or the Late Arrival model, cf. [3]. We focus on the case of balanced load, where arriving packets are destined for any particular of the N output ports with equal probabilities of 1/N. Also, it is assumed that packet arrivals at the M inputs of the switch are governed by independent Bernoulli processes. To be consistent with [1], we place emphasis on identical processes of rate λ. It follows that the number of packet arrivals in an arbitrary time slot, denoted by the random variable A has the binomial probabilities M i ai = Pr {A = i} = λ (1 − λ) M−i , i i = 0, 1, . . . , M.
(1)
Notice that the considered switch can be seen as N batch arrival Geo[X] /D/1/1 queuing systems or multiplexing units working in parallel, where the capacity (represented by the server units, and used only during the serving process) is limited to one packet per output. Let us define the random variable D as the number of packets served by the switch during a given time-slot. In order to derive the probabilities dk of k = 0, 1, . . . , N packets being served, we divide the arriving batch of i packets into k subsets of those packets, which are destined for the same outputs. Since an optimum service discipline is supposed the number of packets being served simultaneously is k. The number of ways such a set of i elements can be partitioned into k disjoint, non-empty subsets is 1434-8411/03/57/05-351 $15.00/0
352 K.-D. Langer: Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches given by the Stirling numbers of the second kind denoted by ki . Taking into account the number of ways these k groups of different requests can be assigned to the N servers of the switch, the total set of different arrival patterns consisting of i packets, that are destined for k outputs is given by ki Nk k!. Each of the N i distinct batches that can be generated by a fixed set of i independent and memoryless traffic sources occurs with equal probability. It follows that the probabilities dk of k packets being served in an arbitrary time-slot are ai i N dk = Pr{D = k} = k! , Ni k k i≥k k = 0, 1, . . . , N.
(2)
If the number of arrivals follows the binomial distribution according to (1), we obtain dk = k!
M i i M λ N (1 − λ)M−i , k N k i=0 i k = 0, 1, . . . , min {M, N}. (3)
Finally we assume a very large number of traffic sources (M → ∞) with very low probabilities of generating requests (λ → 0), where the expected number of arrivals E[A] = λM is a finite value. This is the limiting case where the binomial distribution (1) approaches the Poisson distribution with parameter λM. Thus, the size of arriving batches A has the Poisson probabilities ai = (λM) i e−λM/ i!. Additionally, we can apply the exponential generating function of the Stirling numbers of the i i second kind (ex − 1)k / k! = ∞ i=0 k x / i! to (2), which then simplifies to N λM/N k −λM lim dk = e −1 e , k = 0, 1, . . . , N. k M→∞ λ→0
(4) A verification of these results is given in the Appendix.
3. Discussion of the general result Notice the generality of the closed-form expression given by (2); it holds for any N ∈ N+ , and it applies to any discrete probability mass function of the arrival batch-size. Even it holds for any k ∈ N, because the definitions of both the arrival batch-size probabilities and the binomial numbers ensure that dk = 0 if k > min{M, N}. It can be shown easily that, if k = 0, then (2) simplifies to d0 = a0 , because there are no departures only if there are no requests. As an example, Fig. 1 illustrates the probability distribution of the arrival and the departure batch-sizes according to (1) and (3) respectively, in case of a 32 × 32
Fig. 1. Discrete batch-size distribution of arriving packets (thin lines) and of departures (thick lines) in case of a 32 × 32 switch at low, medium and high utilisation.
switch (M = N = 32) under different input loads. Because the only reason for differences at a given load is the loss of packets due to output contention, in general mostly affected is the upper region of the batch-size. For the same reason, there are only little differences in the batch-size probabilities at the inputs and the outputs of the switch at low utilisation. With growing input loads, the effect of packet losses becomes visible more clearly. Most differences in the batch-size of arrivals and departures occur at heavy loads (i. e. at input saturation), where the arrival batch constantly consists of M requests. The packet loss probability also depends on the ratio of M and N. Choosing M < N results in smaller packet losses due to a lower output utilisation compared to cases where M ≥ N. Thus, the departure batch-size probabilities then move towards values of the arrival batch-size. Usually with sufficient precision, the Poisson distribution can replace a binomial distribution in a range of λ ≤ 0.08 , M ≥ 1500 λ. This fact also applies to (4), because very low traffic loads yield di ≈ ai , as we have discussed above. At higher loads, the given maximum of about λ = 0.08 implies M ≥ 120, and thus the difference between the infinite summation inherent in (4) and the finite summation of (3) becomes negligible. Thus it appears that (4) excellently approximates the departure batch-size probabilities given by (3) within the aforementioned range. Naturally, it provides exact results if the arriving batches are from an infinite number of sources, where the probability of generating a request tends to zero (or if the arrival batch-size has Poisson distribution, e. g. simply with parameter 0 ≥ λ p ≥ 1, respectively). When considering saturated input ports (i. e. a M = 1), or if we assume more generally batches of fixed size i arriving permanently, (2) simplifies to
1 dk a =1 = i i N
i N k! , k k
i = 1, 2, . . . , M,
(5)
K.-D. Langer: Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches 353
Acknowledgement. The author would like to thank the anonymous reviewers for their valuable comments.
Appendix A.1 Verification of the general result In addition to the discussion above, we further verify the departure batch-size probabilities obtained from (2)–(4) in this section. First, when adding up the probabilities dk of (2), we get N N ∞ ai i N k! . (A1) dk = i k N k=0 k k=0 i=0 Fig. 2. Discrete probability distribution of the packet batch-size at the output ports of several saturated N × N crossbars (batch arrivals of fixed size N at λ → 1).
since a j = 0 if j = i. The impact of the switch-size on the departure batchsize distribution under input saturation (i = M) according to (5) is shown in Fig. 2. As can be seen, the shape of the graphs for switches with different finite size is very similar, and clearly the range of the departure batchsize (in the general case of a M × N switch) is 1 ≤ k ≤ min{M, N}, while the level of probabilities decreases towards arbitrary small values with growing size of the switch. Certainly, the results achieved are not restricted to the field of packet switching, as the only constraint of (2)–(5) is that the customers competing for service must come from independent memoryless sources, and that the servers are utilised homogeneously. The arbitration unit must simply ensure that each server starts service, if there are one or more dedicated requests.
4. Conclusion We have examined the departure process of unbuffered M × N packet switches based on rearrangeable or strictly non-blocking interconnection networks, assuming homogeneous utilisation of the outgoing links and an optimum arbitration regime. An exact closed-form solution for the number of busy outputs, i. e. the discrete probability distribution of the packet departure batch-size has been derived and its validity has been proven. The solution is easy to use and it yields high-quality results even at levels of very small probabilities. It applies to any kind of queuing system where batches of requests are independent and of general distributed size when arriving at the servers. Consequently, the solution presented is of relevance to a wide range of multiple client multiple server systems beyond the subject of packet switching.
By applying the combinatorial relationship n x k! xn = k k
(A2)
k≥0
we obtain, as expected, N
dk =
k=0
∞ ai i N =1. Ni i=0
(A3)
Moreover, as an alternative to the approach presented in [1], we can derive the throughput of an unbuffered crossbar under balanced load from the expected number of packets served per time-slot by means of the relation N k dk/N. Thus by applying (3), we get γ = k=0 N M 1 N k i M i γ= k! i λ (1 − λ) M−i N k N k i k=0 i=0
λ M =1− 1− . N
(A4)
Proof: After expanding the expression on the right of (A4) to
λ M 1− 1− = [λ + (1 − λ)] M N
M λ − λ− + (1 − λ) , (A5) N we can rewrite (A5) as the difference of two finite sums by means of the binomial formula. This yields M M i λ (1 − λ) M−i i i=0 M
M λ i − λ− (1 − λ) M−i i N i=0 M i M λ = (1 − λ)M−i N i −(N − 1)i . (A6) i N i=0
354 K.-D. Langer: Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches By applying the combinatorial relationship (A2) to the last term of (A6), this expression again can be expanded to a difference of finite sums. After some algebraic manipulations we obtain i N k k k=0
i i N N −k − k! k k N k=0 i i N 1 = k! k. k N k=0 k
N i − (N − 1)i =
i
k!
the throughput by means of (4): N N k 1 1 N λM/N γ= k dk = k e − 1 e−λM N k=0 N k=0 k N k N − 1 λM/N = e−λM e −1 k−1 k=1 N−1 k N − 1 λM/N = e−λM eλM/N − 1 e −1 k k=0 N−1 = e−λM eλM/N − 1 eλM/N = 1 − e−λM/N . (A8)
(A7)
Since the parameters i and N both limit the summation by definition of the Stirling numbers of the second kind and the binomial numbers respectively, the upper limit of this summation can be exchanged for N. After modifying (A7) in that way and substitution into (A6) we can confirm the identity of (A4). If the number of inputs M becomes very large and the distribution of the arrival batch-size is approximated by the Poisson distribution with parameter λM, we can derive
Again, this is consistent with the result presented in [1].
References [1] Patel, J. H.: Performance of processor-memory interconnections for multiprocessors. IEEE Trans. Comp., c-30 (1981), 771–780. [2] Kolias, C.: Analysis and performance evaluation of new architectures in high-speed packet switching. University of California, Computer Science Department, Los Angeles, 1999. [3] Takagi, H.: Queueing Analysis, Vol. 3: Discrete-Time Systems. Elsevier, Amsterdam, 1993.
Letter Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches Klaus-Dieter Langer Abstract: In this letter, we present an exact closed-form solution for the probability distribution of the packet batch-size at the output ports of unbuffered crossbar based packet switches. The general case of a switch with M inputs and N outputs is considered under balanced traffic and using an optimum arbitration scheme. The solution can be applied to a wide range of queuing systems, even beyond the field of packet switching. Keywords: Unbuffered crossbar, Packet switching, ATM, Discretetime analysis, Batch-size distribution
1. Introduction We consider a unicast packet switch with M inputs and N outputs, where incoming and outgoing links are assumed to run at the same speed. Switching is performed by a nonblocking M × N interconnection network; throughout this paper, we simply assume a crossbar. Furthermore, we consider the special case where there are no buffers for storing packets inside the system, and where from each arriving batch a maximum number of packets will be selected for switching by an appropriate arbitration unit. This is done on basis of random decisions, if the batch contains more than one packet destined for identical outputs. Requests that are not accepted due to output contention (i. e. external blocking) are dropped. Architectures like this are suitable e. g. for interconnecting multiple processors to shared memory modules, or for photonic switching, where buffering of packets in the optical domain (via delay lines) or moving them to electronics for buffering is avoided for the sake of simplicity. Performance measures of this very fundamental system, which is clearly the simplest case of a packet switch, represent the lower bounds for numerous more sophisticated switches such as input-buffered or output-buffered crossbars. The throughput of an unbuffered crossbar switch in case of balanced traffic load and optimum service, derived by Patel in his well-known paper [1], is γ = 1 − (1 − λ/N) M , {M, N} ∈ N+ . For switches with a large number of ports, the throughput converges to lim M,N→∞ γ = 1 − e−λM/N . Apparently, a closed-form solution for the probability distribution of the departing
Received November 25, 2002. Revised Februar 18, 2003. K.-D. Langer, Fraunhofer Institut für Nachrichtentechnik, HeinrichHertz-Institut, Department of Optical Networks, Einsteinufer 37, D-10587 Berlin, Germany. E-mail: [email protected]. de ¨ 57 (2003) No. 5, 351−354 Int. J. Electron. Commun. (AEU)
packet batch-size is not known in literature; an approach for deriving recursively the output port occupancy distribution can be found in [2]. This letter provides an exact and easy to apply closed-form solution based on the Stirling numbers of the second kind. The paper is organised as follows. First, we introduce the traffic model, and then we analyse the batch-size of packets leaving the switch. In addition to the corresponding discussion, a verification of the main result is given in the Appendix.
2. Assumptions and analysis The following assumptions are the same as used in [1]. We consider a discrete-time model with M inputs and N servers (output ports). Time is divided into time-slots of constant length corresponding to the transmission time of the fixed-sized packets to be switched. We suppose synchronous operation, where all requests arrive at time-slot boundaries. Since the service of a packet, i. e. the transfer from the input to the desired output port, takes exactly one time-slot, it follows that a packet, if it was accepted by the switch, has left the system at the time-slot boundary next to the arrival instant. Notice that the analysis is not bound to a specific discrete-time model such as the Early Arrival or the Late Arrival model, cf. [3]. We focus on the case of balanced load, where arriving packets are destined for any particular of the N output ports with equal probabilities of 1/N. Also, it is assumed that packet arrivals at the M inputs of the switch are governed by independent Bernoulli processes. To be consistent with [1], we place emphasis on identical processes of rate λ. It follows that the number of packet arrivals in an arbitrary time slot, denoted by the random variable A has the binomial probabilities M i ai = Pr {A = i} = λ (1 − λ) M−i , i i = 0, 1, . . . , M.
(1)
Notice that the considered switch can be seen as N batch arrival Geo[X] /D/1/1 queuing systems or multiplexing units working in parallel, where the capacity (represented by the server units, and used only during the serving process) is limited to one packet per output. Let us define the random variable D as the number of packets served by the switch during a given time-slot. In order to derive the probabilities dk of k = 0, 1, . . . , N packets being served, we divide the arriving batch of i packets into k subsets of those packets, which are destined for the same outputs. Since an optimum service discipline is supposed the number of packets being served simultaneously is k. The number of ways such a set of i elements can be partitioned into k disjoint, non-empty subsets is 1434-8411/03/57/05-351 $15.00/0
352 K.-D. Langer: Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches given by the Stirling numbers of the second kind denoted by ki . Taking into account the number of ways these k groups of different requests can be assigned to the N servers of the switch, the total set of different arrival patterns consisting of i packets, that are destined for k outputs is given by ki Nk k!. Each of the N i distinct batches that can be generated by a fixed set of i independent and memoryless traffic sources occurs with equal probability. It follows that the probabilities dk of k packets being served in an arbitrary time-slot are ai i N dk = Pr{D = k} = k! , Ni k k i≥k k = 0, 1, . . . , N.
(2)
If the number of arrivals follows the binomial distribution according to (1), we obtain dk = k!
M i i M λ N (1 − λ)M−i , k N k i=0 i k = 0, 1, . . . , min {M, N}. (3)
Finally we assume a very large number of traffic sources (M → ∞) with very low probabilities of generating requests (λ → 0), where the expected number of arrivals E[A] = λM is a finite value. This is the limiting case where the binomial distribution (1) approaches the Poisson distribution with parameter λM. Thus, the size of arriving batches A has the Poisson probabilities ai = (λM) i e−λM/ i!. Additionally, we can apply the exponential generating function of the Stirling numbers of the i i second kind (ex − 1)k / k! = ∞ i=0 k x / i! to (2), which then simplifies to N λM/N k −λM lim dk = e −1 e , k = 0, 1, . . . , N. k M→∞ λ→0
(4) A verification of these results is given in the Appendix.
3. Discussion of the general result Notice the generality of the closed-form expression given by (2); it holds for any N ∈ N+ , and it applies to any discrete probability mass function of the arrival batch-size. Even it holds for any k ∈ N, because the definitions of both the arrival batch-size probabilities and the binomial numbers ensure that dk = 0 if k > min{M, N}. It can be shown easily that, if k = 0, then (2) simplifies to d0 = a0 , because there are no departures only if there are no requests. As an example, Fig. 1 illustrates the probability distribution of the arrival and the departure batch-sizes according to (1) and (3) respectively, in case of a 32 × 32
Fig. 1. Discrete batch-size distribution of arriving packets (thin lines) and of departures (thick lines) in case of a 32 × 32 switch at low, medium and high utilisation.
switch (M = N = 32) under different input loads. Because the only reason for differences at a given load is the loss of packets due to output contention, in general mostly affected is the upper region of the batch-size. For the same reason, there are only little differences in the batch-size probabilities at the inputs and the outputs of the switch at low utilisation. With growing input loads, the effect of packet losses becomes visible more clearly. Most differences in the batch-size of arrivals and departures occur at heavy loads (i. e. at input saturation), where the arrival batch constantly consists of M requests. The packet loss probability also depends on the ratio of M and N. Choosing M < N results in smaller packet losses due to a lower output utilisation compared to cases where M ≥ N. Thus, the departure batch-size probabilities then move towards values of the arrival batch-size. Usually with sufficient precision, the Poisson distribution can replace a binomial distribution in a range of λ ≤ 0.08 , M ≥ 1500 λ. This fact also applies to (4), because very low traffic loads yield di ≈ ai , as we have discussed above. At higher loads, the given maximum of about λ = 0.08 implies M ≥ 120, and thus the difference between the infinite summation inherent in (4) and the finite summation of (3) becomes negligible. Thus it appears that (4) excellently approximates the departure batch-size probabilities given by (3) within the aforementioned range. Naturally, it provides exact results if the arriving batches are from an infinite number of sources, where the probability of generating a request tends to zero (or if the arrival batch-size has Poisson distribution, e. g. simply with parameter 0 ≥ λ p ≥ 1, respectively). When considering saturated input ports (i. e. a M = 1), or if we assume more generally batches of fixed size i arriving permanently, (2) simplifies to
1 dk a =1 = i i N
i N k! , k k
i = 1, 2, . . . , M,
(5)
K.-D. Langer: Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches 353
Acknowledgement. The author would like to thank the anonymous reviewers for their valuable comments.
Appendix A.1 Verification of the general result In addition to the discussion above, we further verify the departure batch-size probabilities obtained from (2)–(4) in this section. First, when adding up the probabilities dk of (2), we get N N ∞ ai i N k! . (A1) dk = i k N k=0 k k=0 i=0 Fig. 2. Discrete probability distribution of the packet batch-size at the output ports of several saturated N × N crossbars (batch arrivals of fixed size N at λ → 1).
since a j = 0 if j = i. The impact of the switch-size on the departure batchsize distribution under input saturation (i = M) according to (5) is shown in Fig. 2. As can be seen, the shape of the graphs for switches with different finite size is very similar, and clearly the range of the departure batchsize (in the general case of a M × N switch) is 1 ≤ k ≤ min{M, N}, while the level of probabilities decreases towards arbitrary small values with growing size of the switch. Certainly, the results achieved are not restricted to the field of packet switching, as the only constraint of (2)–(5) is that the customers competing for service must come from independent memoryless sources, and that the servers are utilised homogeneously. The arbitration unit must simply ensure that each server starts service, if there are one or more dedicated requests.
4. Conclusion We have examined the departure process of unbuffered M × N packet switches based on rearrangeable or strictly non-blocking interconnection networks, assuming homogeneous utilisation of the outgoing links and an optimum arbitration regime. An exact closed-form solution for the number of busy outputs, i. e. the discrete probability distribution of the packet departure batch-size has been derived and its validity has been proven. The solution is easy to use and it yields high-quality results even at levels of very small probabilities. It applies to any kind of queuing system where batches of requests are independent and of general distributed size when arriving at the servers. Consequently, the solution presented is of relevance to a wide range of multiple client multiple server systems beyond the subject of packet switching.
By applying the combinatorial relationship n x k! xn = k k
(A2)
k≥0
we obtain, as expected, N
dk =
k=0
∞ ai i N =1. Ni i=0
(A3)
Moreover, as an alternative to the approach presented in [1], we can derive the throughput of an unbuffered crossbar under balanced load from the expected number of packets served per time-slot by means of the relation N k dk/N. Thus by applying (3), we get γ = k=0 N M 1 N k i M i γ= k! i λ (1 − λ) M−i N k N k i k=0 i=0
λ M =1− 1− . N
(A4)
Proof: After expanding the expression on the right of (A4) to
λ M 1− 1− = [λ + (1 − λ)] M N
M λ − λ− + (1 − λ) , (A5) N we can rewrite (A5) as the difference of two finite sums by means of the binomial formula. This yields M M i λ (1 − λ) M−i i i=0 M
M λ i − λ− (1 − λ) M−i i N i=0 M i M λ = (1 − λ)M−i N i −(N − 1)i . (A6) i N i=0
354 K.-D. Langer: Departure Batch-size Distribution of Unbuffered Crossbar Packet Switches By applying the combinatorial relationship (A2) to the last term of (A6), this expression again can be expanded to a difference of finite sums. After some algebraic manipulations we obtain i N k k k=0
i i N N −k − k! k k N k=0 i i N 1 = k! k. k N k=0 k
N i − (N − 1)i =
i
k!
the throughput by means of (4): N N k 1 1 N λM/N γ= k dk = k e − 1 e−λM N k=0 N k=0 k N k N − 1 λM/N = e−λM e −1 k−1 k=1 N−1 k N − 1 λM/N = e−λM eλM/N − 1 e −1 k k=0 N−1 = e−λM eλM/N − 1 eλM/N = 1 − e−λM/N . (A8)
(A7)
Since the parameters i and N both limit the summation by definition of the Stirling numbers of the second kind and the binomial numbers respectively, the upper limit of this summation can be exchanged for N. After modifying (A7) in that way and substitution into (A6) we can confirm the identity of (A4). If the number of inputs M becomes very large and the distribution of the arrival batch-size is approximated by the Poisson distribution with parameter λM, we can derive
Again, this is consistent with the result presented in [1].
References [1] Patel, J. H.: Performance of processor-memory interconnections for multiprocessors. IEEE Trans. Comp., c-30 (1981), 771–780. [2] Kolias, C.: Analysis and performance evaluation of new architectures in high-speed packet switching. University of California, Computer Science Department, Los Angeles, 1999. [3] Takagi, H.: Queueing Analysis, Vol. 3: Discrete-Time Systems. Elsevier, Amsterdam, 1993.