photonic-ATM-perf

photonic-ATM-perf - Performance Analysis of a Photonic...

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Unformatted text preview: Performance Analysis of a Photonic Single-Hop ATM Switch Architecture with Bursty and Correlated Arrivals Martin W. McKinnon George N. Rouskas Harry G. Perros CACC TR-96 25 June 1996 Abstract We consider a photonic ATM switch based on the single-hop WDM architecture. The switch operates under schedules that mask the transceiver tuning latency. We develop an exact queueingbased decomposition algorithm to obtain the queue-length distribution at the input and output ports of the switch. The analysis is carried out using arrival models that capture the notion of burstiness and correlation, two important characteristics of ATM type of tra c, and which permit non-uniform destinations. We then derive analytic expressions for the cell-loss probability at the input and output ports, and the delay distribution to traverse the switch. To the best of our knowledge, such a comprehensive performance analysis of an optical switch architecture has not been done before. Keywords: Optical networks, performance evaluation, photonic ATM switch, Markov modulated Bernoulli process MMBP, Wavelength Division Multiplexing WDM Center for Advanced Computing and Communication North Carolina State University Raleigh, NC 27695 1 Introduction One of the issues in Asynchronous Transfer Mode ATM networks is that of developing switch architectures that can e ectively switch cells at very high data rates currently, data rates on the order of a few tens of Gigabits per second per port are envisioned. Over the last decade, a great deal of research e ort has been devoted to the design of fast cell switches suitable to a broadband, integrated services environment; surveys of some of these architectures may be found in 1, 2 . Mainstream research and development activities in the area of broadband switching are focused exclusively on electronics-based technologies which have attained a high level of maturity. On the other hand, the deployment of optics is limited to mere point-to-point transmission where the technology has proven successful in a short time span. Given the continued rapid progress in lightwave technology including the demonstration of fast tunable transceivers 3, 4 , the development of erbium-doped ber ampli ers 5 , and guided-wave optical switching 6 , and the anticipated total dominance of optical ber in the wired network, the issue of deeper penetration of optics naturally arises. Given the potential of optical solutions to cell switching, the possibility of employing photonics to implement switching functions hitherto reserved for electronics is currently being explored see 7 and references thereof. However, there remain at least two major technical challenges to be overcome before one can contemplate the design of all-optical switches. First, there is the di culty of controlling light by light", and secondly, the technologies for implementing bu ering in the optical domain are not yet mature enough. Consequently, the most likely scenarios for near-term photonic cell switching will involve an optical switching fabric with electronic control and bu ering. It has long been recognized that Wavelength Division Multiplexing WDM will be instrumental in bridging the gap between the speed of electronics and the virtually unlimited bandwidth available within the optical medium. The wavelength domain adds a signi cant new degree of freedom to network design, allowing new network concepts to be developed. With a few exceptions e.g., 8, 9, 10 , however, most broadcast WDM architectures that have appeared in the literature require a large number of wavelengths and or very fast tunable transceivers 11 - 12 . Furthermore, the performance analysis of these architectures has been typically carried out assuming uniform tra c and memoryless arrival processes see most of the above references, as well as 13 - 14 . Similar tra c assumptions were made in performance studies of electronic ATM switches in the late 80s. However, as it was later shown, such assumptions lead to overwhelmingly erroneous results regarding the performance of a switch. In order to study correctly the performance of a switch, one needs to use tra c models that capture the notion of burstiness and correlation, and which permit 1 non-uniform output port destinations 15 . In this paper we revisit the well known and widely studied single-hop broadcast-and-select WDM architecture 16 . Unlike previous papers, however, we develop an exact queueing-based decomposition algorithm to study the performance of a single-hop ATM switch architecture operating under schedules that mask the transceiver tuning latency 9 . The analysis is carried out using arrival models that capture the important characteristics of ATM type of tra c, and nonuniform destinations. To the best of our knowledge, such a comprehensive performance analysis of an optical switch architecture has not been done before. The next section presents our system model and provides some background information. The performance analysis of the switch is presented in Sections 3 and 4, and we conclude the paper in Section 5. 2 The ATM Switch Under Study 2.1 The Switch Architecture We consider an optical switch architecture with N input ports and N output ports interconnected through a broadcast passive star the switch fabric that can support C  N wavelengths 1;    ; C see Figure 1. Each input port is equipped with a laser that enables it to inject signals into the optical medium. Similarly, each output port is capable of receiving optical signals through an optical lter. The laser at each input port is assumed to be tunable over all available wavelengths. The optical lters, on the other hand, are xed to a given wavelength. Let j  denote the receive wavelength of output port j . Since C  N , a set Rc of output ports may be sharing a single receive wavelength c : Rc = fj j j  = cg; c = 1;    ; C 1 The switch operates in a slotted mode. Since there are N ports but C  N channels, each channel must run at a rate N times faster than the rate of the input or output links  N need C C not be an integer. Thus, we distinguish between arrival slots which correspond to the ATM cell transmission time at the input output link rate and service slots which are equal to the cell transmission time at the channel rate within the switch. Obviously, the duration of a service slot C is equal to N times that of an arrival slot. Without loss of generality, we assume that all input links are synchronized at arrival slot boundaries; similarly for output links. On the other hand, all C channels internal to the switch are synchronized at service slot boundaries. 2 input queues 1 input port 1 C tunable lasers fixed optical filters output queues output port 1 . . . . . . λ 1 ... λ C λ 1 ... λ C λ(1) . . . output port N passive star 1 input port N C . . . λ(N) λ λ 1 ... λ C ... λ C 1 Figure 1: Queueing model of a switch architecture with N ports and C wavelengths The switch employs electronic queueing at both the input and output ports, as Figure 1 illustrates. Cells arrive at an input port i and are bu ered at a nite capacity queue, if the queue is not full. Otherwise, they are dropped. As Figure 1 indicates, the bu er space at each input port is assumed to be partitioned into C independent queues. Each queue c at input port i contains cells destined for the output ports which listen to a particular wavelength c ; c = 1;    ; C . This arrangement eliminates the head-of-line problem, and permits an input port to send a number of  cells back-to-back when tuned to a certain wavelength. We let Bicin denote the capacity of the queue at input port i corresponding to wavelength c. Cells bu ered at an input port are transmitted on a FIFO basis onto the optical medium by the port's laser. This transmission takes place on an appropriate service slot which guarantees that the cell will be correctly received by its destination output port more on this in the next subsection. Upon arriving at the output port, the cell is once again placed at a nite capacity bu er. Let Bjout denote the bu er capacity of output port j . Cells arriving at an output port to nd a full bu er are lost. Cells in an output bu er are also served on a FIFO basis. Interest in such a photonic switch architecture arises from several observations: it is highly modular, allowing the switch to grow relatively easily by adding ports and wavelengths; it is scalable, since the number of wavelengths need not be equal to the number of ports, and since the data rate within the switch needs only be N times the rate of the input output links; C it provides end-to-end optical paths; 3 its hardware requirements, in terms of the number of transceivers per port is minimum; it can be recon gured to adapt to changing tra c patterns or to overcome failures of ports or transceivers; and it does not require extremely fast tunable transmitters as explained below, and thus can be built using currently available tunable optical devices. 2.2 Transmission Schedules One of the potentially di cult issues that arise in a WDM environment, such as the one described above, is that of coordinating the various transmitters receivers. Some form of coordination is necessary because a a transmitter and a receiver must both be tuned to the same channel for the duration of a cell's transmission, and b a simultaneous transmission by one or more input ports on the same channel will result in a collision. The issue of coordination is further complicated by the fact that tunable transceivers need a non-negligible amount of time to switch between wavelengths. For the Gigabit per second rates envisioned here, and for 53-byte ATM cells, the tuning latency of state-of-the-art tunable lasers or lters can be as long as several times the size of a service slot 3 . Consequently, approaches that require each tunable transmitter to send a single cell and then switch to a new wavelength, will su er a high tuning overhead and will result in a very low throughput. In a recent paper 9 it was shown that careful scheduling can mask the e ects of arbitrarily long tuning latencies, making it possible to build high-throughput photonic ATM switches using currently available lightwave technology. The key idea is to have each tunable transmitter send a block of cells on a wavelength before switching to another one. The main result of 9 was a set of new algorithms for constructing near-optimal and, under certain conditions, optimal schedules for transmitting a set of tra c demands faic g. Quantity aic represents the number of cells to be transmitted by input port i onto channel c per frame. The schedules are such that no collisions ever occur. Furthermore, they are easy to implement in a high speed environment, since the order in which the various input ports transmit is the same for all channels 9 . Quantity aic ; i = 1;    ; N; c = 1;    ; C , can be seen as the number of service slots per frame allocated to input port i, so that the port can satisfy the required QoS of its incoming tra c intended for wavelength c . By xing aic , we indirectly allocate a certain amount of the bandwidth of wavelength c to port i. This bandwidth could be an upper bound of the e ective bandwidth of the total tra c carried by input port i on wavelength c. In general, the estimation of the quantities aic ; i = 1;    ; N; c = 1;    ; C; is part of the call admission algorithms, and is beyond the scope of this paper. We note that as the tra c varies, aic may vary as well. In this paper, 4 Frame a1 c λc 0 1 2 g1 c a2 c g2 c aNc gN c M-1 (a) a2 c g2 c . . . arrival slot (b) service slot . . . Figure 2: a Schedule for channel c , and b detail corresponding to input port 2 we assume that quantities aic are xed, since this variation will more likely take place over larger scales in time. We assume that transmissions by the input ports onto wavelength c follow a schedule as shown in Figure 2. This schedule repeats over time. Each frame of the schedule consists of M arrival slots. Within each frame, input port i is assigned aic contiguous service slots for transmitting cells on channel c . These aic slots are followed by a gap of gic  0 slots during which no port can transmit on c . This gap may be necessary to ensure that input port i + 1 has su cient time to tune from wavelength c,1 to c before it starts transmission. The algorithms in 9 are such that the number of slots in most of the gaps is equal to either zero or a small integer. Thus, the P length of the schedule is very close to the lower bound maxi f C=1 aic g. Note that in Figure 2 we c have assumed that an arrival slot is an integer multiple of service slots, but this need not be true in general, and it is not a necessary assumption for our model. Observe also that, although the frame begins and ends on arrival slot boundaries, the beginning or end of transmissions by a port does not necessarily coincide with the beginning or end of an arrival slot although it is, obviously, synchronized with service slots. 2.3 Tra c Model The arrival process to each input port of the switch is characterized by a two-state Markov Modulated Bernoulli Process MMBP, hereafter referred to as 2-MMBP. This is a Bernoulli process 5 whose arrival rate varies according to a two-state Markov chain. It captures the notion of burstiness and the correlation of successive interarrival times, two important characteristics of ATM type of tra c. For details on the properties of the 2-MMBP, the reader is referred to 17 . We note that the algorithm for analyzing the switch was developed so that it can be readily extended to MMBPs with more than two states. We assume that the arrival process to port i; i = 1;    ; N , is given by a 2-MMBP characterized by the transition probability matrix Qi , and by Ai as follows: 2 00 01 3 Qi = 4 qi10 qi11 5 qi qi and 2 Ai = 4 0 0 i 0 5 1 i 3 2 In 2, qikl ; k; l = 0; 1; is the probability that the 2-MMBP will make a transition to state l, given that it is currently at state k. Obviously, qik0 + qik1 = 1; k = 0; 1. Also, a0 and a1 are i i the arrival rates of the Bernoulli process at states 0 and 1, respectively. Transitions between states of the 2-MMBP occur only at the boundaries of arrival slots. We assume that the arrival process to each input port is given by a di erent 2-MMBP. Let rij denote the probability that a cell arriving to input port i will have j as its destination output port. We will refer to frij g as the routing probabilities; this description implies that the routing probabilities can be input port dependent and non-uniformly distributed. The destination probabilities of successive cells are not correlated. That is, in an input port, the destination of one cell does not a ect the destination of the cell behind it. This is a reasonable assumption when the switch is used as part of a backbone network. Given these assumptions, the probability that a cell arriving to port i will have to be transmitted on wavelength c is: ric = X j 2Rc rij ; i = 1;    ; N 3 3 Exact Queueing Analysis In this section we analyze exactly the queueing network shown in Figure 1. This queueing network represents the tunable-transmitter, xed-receiver switch under study. The arrival process to each input port is assumed to be a 2-MMBP, and the access of the input ports to the wavelengths is governed by the schedule described in Section 2.2. The objective of the analysis of this queueing network is to obtain the queue-length distribution in an input or output port, from which performance measures such as cell-loss probability and delay distribution can be obtained. As will be seen below, the analysis of this queueing network is exact. 6 input queues corresponding to λc 1 λc fixed optical filters output queues listening to λc λc . . . λc N . . . λ c passive star Figure 3: Queueing sub-network for wavelength c We rst note that the original queueing network can be decomposed into C sub-networks, one per wavelength, as in Figure 3. For each wavelength c , the corresponding sub-system consists of N input queues, and all the output queues that listen to wavelength c . Each input queue i of the sub-system, is the one associated with wavelength c in the i-th input port. These input queues will transmit to the output queues of the sub-system over wavelength c. Note that, due to the independence among the C queues at each input port, the transmission schedule, and the fact that each output port listens to a speci c wavelength, this is decomposition is exact. In view of this decomposition, it su ces to analyze a single sub-network, since the same analysis can be applied to all other sub-networks. Consider now the sub-network for wavelength c . We will analyze this sub-network by decomposing it into individual input and output ports. The decomposition algorithm gives an exact solution. As discussed in the previous section, each input queue i of the sub-network is only served for aic consecutive service slots per frame. During that time, no other input port is served. Input queue i is not served in the remaining slots of the frame. In view of this, there is no dependence among the input queues of the sub-system, and consequently each one can be analyzed in isolation in order to obtain its queue-length distribution. Each output port will also be considered in isolation. However, in order to consider each output port in isolation, we need to characterize the departure process from the input queues, which is o ered as the arrival process to the output queues. This characterization is the most critical component of the decomposition algorithm, and is discussed in detail in Section 3.2. From the queueing point of view, the above queueing network can be seen as a polling system in discrete time. Despite the fact that polling systems have been extensively analyzed, we note that very little work has been done within the context of discrete time see for example 18 . In 7 addition, this particular problem di ers from the typical polling system since we consider output queues, not typically analyzed in polling systems. We now proceed to obtain the queue-length distribution at an input queue. Following that, we obtain the queue-length distribution at an output queue. 3.1 The Queue-Length Distribution of an Input Queue We now turn our attention to the sub-network for wavelength c . The i-th input queue of this sub-network is the c-th queue of input port i. Since throughout this section we only consider the sub-network corresponding to c , we will simply refer to this queue as input queue i". Consider input queue i of the sub-network in isolation. This input queue receives exactly aic service slots on wavelength c , as shown in Figure 4a. The block of aic service slots may not be aligned with the boundaries of the arrival slots. For instance, in the example shown in Figure 4a, the block of aic service slots begins at the second service slot of arrival slot x , 1, and it ends at the end of the second service slot in arrival slot x + 1. Here, x , 1, x, and x + 1 represent the arrival slot number within a frame. For each arrival slot, de ne vic x as the number of service slots allocated to input queue i, that lie within arrival slot x 1. Then, in the example in Figure 4a, we have: vic x , 1 = 3; vicx = 4; vicx + 1 = 2, and vic x0  = 0 for all other x0 . Obviously we have, M ,1 X x=0 vic x = aic 4 We analyze input queue i by constructing its underlying Markov chain embedded at arrival slot boundaries. The order of events is as follows. The service i.e., transmission completion of a cell occurs at an instant just before the end of a service slot. An arrival may occur at an instant just before the end of an arrival slot, but after the service completion instant of a service slot whose end is aligned with the end of an arrival slot. The 2-MMBP describing the arrival process to the queue makes a state transition immediately after the arrival instant. Finally, the Markov chain is observed at the boundary of each arrival slot, after the state transition by the 2-MMBP. The order of occurrence of these events is shown in Figure 4b. The state of the input queue is described by the tuple x; y; z , where: vicx In Figure 4 we assume that each arrival slot contains an integral number of service slots. If this is not the case, is de ned as the number of service slots that lie completely within arrival slot x, plus one if there is a service slot that lies partially within slots x , 1 and x. If there is a service slot that lies partially within arrival slots x and x + 1, it will be counted in vic x + 1. 1 8 Frame a ic λc x-1 x (a) x+1 arrival instant 2-MMBP state transition instant service completion instant (b) x+1 observation instant Figure 4: a Service period of input port i on channel c , and b detail showing the relationship among service completion, arrival, 2-MMBP state transition, and observation instants within a service and an arrival slot Current State x; y; z  x; y; z  Table 1: Transition probabilities out of state x; y; z  of the Markov chain x  1; maxf0; y , vic xg; z 0  x  1; minfBicin ; maxf0; y , vic xg + 1g; z 0 Next State Transition Probability qizz 1 , z ric i qizz  z ric i 0 0 x represents the arrival slot number within a frame x = 0; 1;   ; M , 1,  y indicates the number of cells in the input queue y = 0; 1;    ; Bicin, and z indicates the state of the 2-MMBP describing the arrival process to this queue, that is, z = 0; 1. It is straightforward to verify that, as the state of the queue evolves in time, it de nes a Markov chain. Let  denote modulo-M addition, where M is the number of arrival slots per frame. Then, the transition probabilities out of state x; y; z  are given in Table 1. Note that, the next state after x; y; z  always has an arrival slot number equal to x  1. In the rst row of Table 1 we assume that the 2-MMBP makes a transition from state z to state z 0 from 2, this event has a probability qizz  of occurring, and that no cell arrives to this queue during the current slot from 2 and 3, this occurs with probability 1 , z ric. Since at most vic x cells are serviced during this i arrival slot, and since no cell arrives, the queue length at the beginning of the next slot is equal to 0 9 maxf0; y , vic xg. In the second row of Table 1 we assume that the 2-MMBP makes a transition from state z to state z 0 and a cell arrives to the queue. This arriving cell cannot be serviced during this slot, and has to be added to the queue. Finally, the expression for the new queue length ensures  that it will not exceed the capacity Bicin of the input queue. We observe that the probability transition matrix of this Markov chain has the following block form: Sic = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 Ric M , 1 0 0 0 . . . 0 Ric 0 0 0 . . . 0 0 0 Ric 1 0 . . . 0 0 0 0 Ric 2 . . . 0 0      . . . 3 7 0 7 1 7 7 7 2 7 7 . 7 .. 7 7 7 Ric M , 2 7 M , 2 5 0 M ,1 0 0 0 . . . 5 Note that the only non-zero elements of the M  M matrix Sic are those of the upper diagonal. This is due to the fact that at each transition instant, i.e., at each arrival slot boundary, the random variable x changes to x  1. Changes in the other two random variables, y and z , of the state of   the queue are governed by the Bicin + 1  Bicin + 1 matrices Ric x. There are M di erent Ric matrices, one for each arrival slot x in the frame. De ne matrices Xic and Yic as follows: Xic = ric Ai Qi and Yic = I , ric Ai Qi ; 6 where I is the identity matrix. Then, the transition matrix Ric x associated with arrival slot x can be written as: 0  0 7 0 . . . .7 . . . . .7 . . . . .7 . 7 vic x Xic 0  0 7 7 Ricx = 7 7 Yic Xic 0    0 7 vic x + 1 7 7 vic x + 2 0 Yic Xic 0  0 7 . . . . . .7 . . . . . . .7 . . . . . . .7 . 5 in 0 0    0 Yic Xic 0    0 Bic The structure of matrix Ric x given in 7 can be explained as follows. Suppose that the number of cells y in the queue at the beginning of slot x is at most vic x. Since up to vic x cells 0 . . . 0 0 . . . 0 0 0 . . . 0 0 0 . . . 0 . . . 0 0 0 . . . 10 2 6 Y.ic 6 .. 6 6 6 Yic 6 6 6 0 6 6 6 0 6 6 . 6 . 6 . 4 Xic 3 can be served within slot x, the number in the queue at the end of the slot will be 1 or 0, depending on whether an arrival occurred or not. This is indicated by the transitions in rows 0 through vic x of matrix Ric x. However, if at the beginning of slot x we have y vic x, then the number in the queue at the next transition will be y , vic x plus one if an arrival occurred. This is indicated by the transitions in rows vic x + 1 and higher of Ric x. Of course, y cannot exceed the queue  capacity Bicin . Since the number of service slots vic x depends on the particular slot x within the frame, Ric x is a function of x. Matrix Ric x is slightly di erent when vic x = 0. This is because, in this case, if the state of   the input queue is such that y = Bicin , a new arrival will be discarded. So when y = Bicin , the 2-MMBP is allowed to make a transition, but regardless of whether or not an arrival is generated,  the number of cells in the queue will remain equal to Bicin . Thus, the the last row of Ric x will be: 0 0    0 Qi . It is now straightforward to verify that the Markov chain with transition matrix Sic is irreducible, and therefore a steady-state distribution exists. Transition matrix Sic de nes a p-cyclic Markov chain 19 , and therefore it can be solved using any of the techniques for p-cyclic Markov chains in 19, ch. 7 . We have used the power method in 19, ch. 7 to obtain the steady state probability ic x; y; z that at the beginning of arrival slot x, the 2-MMBP is in state z and the input queue has y cells. The steady-state probability that the queue has y cells at the beginning of slot x, independent of the state of the 2-MMBP is: ic x; y = X z=0;1 icx; y; z  8 Finally, we note that all the results obtained in this subsection can be readily extended to MMBP-type of arrival processes with more than two states. For this, it would su ce to appropriately modify matrices Xic and Yic . 3.2 The Queue-Length Distribution of an Output Queue We now proceed to obtain the queue-length distribution of an output queue. Let j be an output port listening to wavelength c . Below, we analyze this queue in isolation. As will be seen, the analysis is exact. In order to analyze queue j in isolation, we need to characterize the arrival process to the queue. This arrival process is, in fact, the departure process from the input queues. An interesting aspect of the departure process from the input queues is that for each frame, during the sub-period aic we only have departures from the i-th input queue. This period is then followed by a gap gic 11 during which no departure occurs. This cycle repeats for the next input queue. Thus, in order to characterize the overall departure process o ered as the arrival process to output queue j , it su ces to characterize the departure process from each input queue, and then combine them together. We note that this overall departure process is quite di erent from the typical process of constructing the superposition of a number of departures into a single stream, where at each slot more than one cell may be departing. As in the previous section, we obtain the queue-length distribution of output port j at arrival slot boundaries. Recall that an arrival slot to an input queue is equal to a departure slot from an output queue. Also, arrival and departure slots are synchronized. Therefore, during an arrival slot x a cell may be transmitted out from the output queue. However, during slot x there may be several arrivals to the output queue from the input queues. Let x; w be the state associated with output port j , where x indicates the arrival slot number within the frame x = 0; 1;    ; M , 1, and w indicates the number of cells at the output queue w = 0; 1;   ; Bjout. We assume the following order of events. A cell will depart from the output queue at an instant immediately after the beginning of an arrival slot and the departure will be completed at the end of the slot. A cell from an input port arrives at an instant just before the end of a service slot. Finally, the state of the queue is observed just before the end of an arrival slot and after the arrival associated with the last service slot in the arrival slot has occurred see Figure 5b. Let uj x be the number of service slots of any input queue on wavelength c within arrival slot x. We have that N X uj x = vicx 9 where vic x is as de ned in 4. Quantity uj x represents the maximum number of cells that may arrive to output port j within slot x. In the example of Figure 5a where we show the arrival slots during which cells from input ports i and i + 1 may arrive to output port j , we have: uj x , 1 = vic x , 1 = 4, uj x = vicx + vi+1;c x = 1 + 2 = 3, and uj x + 1 = vi+1;c x + 1 = 4. Observe now that a at each state transition x advances by one modulo-M , b at most one cell departs from the queue as long as the queue is not empty, c a number s  uj x of cells may be transmitted from the input ports to output port j within arrival slot x, and that d the queue capacity is Bjout . Then, the transition probabilities out of state x; w for this Markov chain are given in Table 2. 12 i=1 Frame a λ ic a i+1,c c x-1 instant at which departure ends x (a) x+1 arrival instant instant at which observation instant (b) x+1 departure starts Figure 5: a Arrivals to output port j from input ports i and i + 1, and b detail showing the relationship of departure, arrival, and observation instants Current State x; w Table 2: Transition probabilities out of state x; w of the Markov chain P Q x  1; minfBjout ; maxf0; w , 1g + sg s ++sN =s N Li si ; x i=1 0  s  uj x 1 Next State Transition Probability Also, de ne ic y j x as the conditional probability of having y cells at the i-th input queue given that we are at the beginning of slot x: ic 11 icy j x = PBin x; y ic ic x; k k=0   In Table 2, Li si ; x is the probability that input port i transmits si cells to output port j in 0 arrival slot x. To compute Li si ; x, de ne rij as the conditional probability that a cell is destined to output port j , given that the cell is destined to be transmitted on c , the receive wavelength of output port j : 0 rij = P rij r = rij 10 r k2Rc ik ic 13 0 Then, for rij 8 : 1, the probability Li si ; x is given by 0; Li si; x = PBicin  y=si  si vic x 0 1 minfy; vicxg A 0 si 0 rij  1 , rij minfy;vic xg,si ; si  vic x icy j x @ si 12 Expression 12 can be explained by noting that input port i will transmit si cells to output port j during arrival slot x if a vic x  si , b input port i has y  si cells in its queue for c at the beginning of the slot, and c exactly si of minfy; vicxg cells that will be transmitted by this queue in this arrival slot are for output j . 0 If rij = 1, in which case j is the only port listening on wavelength c , the expression for Li si ; x must be modi ed as follows: 8 0; si vic x icsi j x; si vic x Li si ; x = 13 in : PB=si icy j x; si = vic x y ic    Expressions 12 and 13 are based on the assumption that vic Bicin and vic Bjout , which we believe is a reasonable one. In the general case, quantity vic in both expressions must be replaced  by minfvic x; Bicin; Bjoutg. The transition matrix Tj of the Markov chain de ned by the evolution of the state x; w of output queue j has the following form, which is similar to that of matrix Sic given by 5: Tj = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 Uj M , 1 0 0 0 . . . 0 Uj 0 0 0 . . . 0 0 0 Uj 1 0 . . . 0 0 0 0 Uj 2 . . . 0 0      . . . 3 7 0 7 1 7 7 7 2 7 7 . 7 .. 7 7 7 Uj M , 2 7 M , 2 5 0 M ,1 0 0 0 . . . 14 the state of the output queue. The elements of this matrix can be determined using Table 2 and expressions 12 and 13. Since they depend on vic x and uj x, Uj x also depends on x, the slot number within the frame. We observe that Tj also de nes a p-cyclic Markov chain. We have used the power method in 19, ch. 7 to obtain j x; w, the steady-state probability that output queue j has w cells at the 14 Uj x is a Bjout + 1  Bjout + 1 matrix that governs changes in random variable w of beginning of slot x. 3.3 Summary of the Decomposition Algorithm Below we summarize our approach to analyzing the sub-network of Figure 3 corresponding to wavelength c . We assume that quantities faic g and the corresponding schedule see 9  are given. 1. For each arrival slot x, use the schedule and expressions 4 and 9 to compute the quantities vic x and uj x, i = 1;    ; N; j : j  = c . 2. For each input queue i, construct the transition probability matrix Sic from 2, 3, 5, 6, and 7. Solve this matrix using any of the techniques in 19, ch. 7 , and use 8 to obtain the steady-state probability ic x; y  that input queue i has y cells at the beginning of the x-th slot of the frame. 3. For each output port j with c as its receive wavelength, use ic x; y  derived in Step 2, and 12-13 to construct the transition matrix Tj given by 14. Solve the matrix as in Step 2 to obtain j x; w, the steady-state probability that port j has w cells in its queue at the beginning of slot x. Note that the complexity of this approach his dominated by Step 2. For each of the N input i h i   queues we have to solve a matrix of dimensions 2M Bicin + 1  2M Bicin + 1 , where M is the  length of the schedule in arrival slots and Bicin is the capacity of the respective queue. Inverting a K  K matrix takes time OK 3, although some of the techniques in 19, ch. 7 can take advantage of the fact that the matrix is sparse to solve for the queue-length distributions at a signi cantly faster rate. Thus, in the worst case, the overall complexity of our algorithm is ONM 3B 3 , where  B = maxi fBicing. 15 4 Cell-Loss Probability and Delay Distribution We now use the queue-length distributions for the input and output ports, ic x; y  and j x; w, respectively, derived in the previous section, to obtain the cell-loss probability and the delay distribution to traverse the switch. 4.1 The Cell-Loss Probability at an Input Port Let ic x be the probability that a cell arriving to the c-th queue of input port i during arrival slot x of the schedule will be lost. Let us also refer to Figure 4b which shows the service completion, arrival, and observation instants within slot x. We observe that, if the number vic x of slots during which this queue is serviced within arrival slot x is not zero i.e., vic x 0, no arriving cell will be lost. This is because at most one cell may arrive in slot x. Even if the c-th queue at input port i is full at the beginning of slot x, vic x  1 cells will be serviced during this slot, and the order of service completion and arrival instants in Figure 4b guarantees that an arriving cell will be accepted. On the other hand, if vic x = 0 for slot x, then an arriving cell will be discarded if and only if the queue is full at the beginning of x. Thus, 0; vic x 0 = : 15 in ; v x = 0 icx 1; Bic ic where denotes regular subtraction with the exception that, if x = 0, then x 1 = M , 1. The probability ic that a cell arriving to the c-th queue of input port i is lost, regardless of the arrival slot x can now be computed from 15 as follows: ic x ic = M ,1 X x=0 8 = X ic x Pr a cell arrives at slot x 16 where the summation runs over all x for which vic x = 0. Note that the state of the arrival process to the i-th input port is independent of the slot number. Thus, the probability that a cell arrives to the c-th queue of input port i, at some slot x, is equal to the steady-state arrival probability for the 2-MMBP times the probability ric that this cell is destined for an output port listening to wavelength c . From 2 and 17 , the steady state arrival probability for this 2-MMBP is 10 0 01 1 = qi ai + qi10ai 17 i qi01 + qi 16 x:vic x=0  ic x 1; Bicin  Pr a cell arrives at slot x Thus, we can rewrite 16 as: ic = X x:vic x=0  ic x 1; Bicin i ric 18 4.2 The Cell-Loss Probability at an Output Port The cell-loss probability at an output port is more complicated to calculate, since we may have multiple cell arrivals to the given output port within a single arrival slot refer to Figure 5a. Let us de ne j x; n as the probability that n cells will be lost at output queue j within arrival slot x. An output port will lose n cells in slot x if a the port had w; 0  w  Bjout cells at the beginning of slot x, b one cell departed during the slot if w 0, and c exactly Bjout , maxf0; w , 1g + n cells arrived during slot x the max operation in this expression takes care of the case w = 0 when no cell may depart. We can then write: j x; n = out BX  j The last probability in 19 can be obtained using 12 and 13: w=0 j x 1; w Pr Bjout , maxf0; w , 1g + n cells arrive in slot x 19 Pr s cells arrive to output port j in slot x = X N Y s1 ++sN =s i=1 Li si ; x 20 Note that at most uj x cells may arrive and get lost in arrival slot x. We can then get the probability j x that a cell is lost given that it arrives to output port j in slot x: E number of arrivals to output port j in slot x The expected number of arrivals in slot x can be computed using 20: E number of arrivals in slot x = uX j x s=1 j x = Puj x n x; n j n=1 21 s Pr s cells arrive in slot x 22 Finally, the probability j that an arriving cell to output port j will be lost regardless of the arrival slot x can be found by unconditioning 21 on the arrival slot. We have: j = M ,1 X x=0 j x Pr a cell arrives to port j in slot x 23 24 This last probability is equal to E x PM ,1 number of arrivals to output port j injslotslot x0 x =0 E number of arrivals to output port in 0 17 cell arrives to find y in queue Frame a ic x x+1 y service slots ∆ ic (x,y) cell arrives to its output port in this service slot Figure 6: De nition of ic x; y  for y aic Thus, we obtain j PM ,1 x E number of arrivals to output port j in slot x = x=0 ,1j PM E number of arrivals to output port j in slot x0 x =0 0 25 4.3 The Delay Distribution In this section we calculate the distribution of the number of arrival slots that elapse from the instant that a cell arrives to an input queue to the instant that the cell departs from an output queue. We note that the cell arrives at, and departs from, the switch on arrival slot boundaries. Let us tag a cell arriving to the c-th queue of input port i in arrival slot x. Let j be the destination output port of this cell, and let c be the wavelength of j . We assume that the tagged  cell sees y cells in the input queue, where y Bicin . De ne ic x; y  to be the number of arrival slots between the end of slot x and the end of the arrival slot during which the tagged cell is transmitted to the output queue j on wavelength c . If y aic , then ic x; y  can be calculated very easily, as shown in Figure 6. Since x and y are given, we can calculate how many arrival slots will elapse until the input queue is served by wavelength c , and subsequently we can calculate the number of arrival slots required to transmit y cells on this wavelength. We note that for y aic we have ic x; y  M , where M is the length of the schedule in arrival slots. Now, let us assume that y  aic . Then, y can be written as y = kaic + y 0 , where k is an integer k  1, and y 0 aic . In this case, we have ic x; y = kM + icx; y 0. Finally, let w, where w Bjout , be the number of cells that the tagged cell will nd in the output queue j upon arrival to this queue. Then, it will wait exactly w arrival slots before it is transmitted out of the switch, for a total of w + 1 slots. 18 We can now compute ij m, the probability that a cell with destination j arriving to port i will spend m arrival slots in the switch as the product of a the probability that the cell will spend l m slots in its input queue, and b the probability that the cell will spend exactly m , l slots in the output queue: ij m = m,1 X l=1 Pr l slots in input queue i Pr m , l slots in output queue j 26 Since the state of the arrival process to input port i is independent of the arrival slot and the number in the queue, using the same reasoning as in 18 we can write: Pr l slots in input queue i = X x;y:ic x;y=l icx; y i rij 27 where the sum is over all states x; y  of the input queue such that the cell will spend exactly l arrival slots in the queue. In order to compute Pr m , l slots in output queue j , let us return to the tagged cell arriving to the c-th queue of input port i in slot x. Suppose that, at the time of its arrival to the switch, its  input and output queues have y and w cells, respectively y Bicin . Then, the amount of time that the tagged cell spends in the switch is a function of x, y , and w. Note that the cell will arrive to its output port in arrival slot x0 = x  ic x; y , where  denotes addition modulo-M refer also to Figure 6. If it nds w0 Bjout cells in its output port at that time, it will spend another w0 + 1 slots in the switch, for a total of icx; y + w0 + 1 slots. To obtain an exact expression for Pr m , l slots in output queue j , we must compute the conditional probability that the cell will nd w0 cells in its output queue in slot x0, given that there were w cells in that same queue in slot x. This conditional probability, however, is di cult to calculate, as it depends on a the schedule, b the occupancy of the c-th queue at all other input ports, and c the routing probabilities. Alternatively, we can make the simplifying assumption that, when a cell is transmitted to its output queue, the probability that it will nd w cells in this queue is equal to the steadyP state probability of having w cells in the queue j w = M ,1 j x; w. This is a reasonable x=0 approximation when a there is a relatively large number of input ports, and b the destination of one cell does not a ect the destination of the cell behind it as we assumed in this paper. Then, we can write: Pr m , l slots in output queue j = j m , l , 1 28 Finally, we can use 27 and 28 to rewrite the probability 26 that a cell will spend m arrival 19 slots in the switch as 9 = ic x; y  i rij j m , l , 1; ij m = : l=1 x;y:ic x;y=l m,1 X 8 X 29 5 Concluding Remarks We have studied the performance of a photonic ATM switch based on the single-hop WDM architecture. We have developed an exact decomposition algorithm to obtained the queue-length distributions at the input and output ports, and we have presented analytic expressions for the cell-loss probability and delay distribution to traverse the switch. Our current research focuses on extending the queueing model presented here in three directions. First, we will consider the scenario where the C queues at each input port are not independent but share a common bu er space. Second, we will introduce backpressure mechanisms to prevent an input port from transmitting a cell to an output port that is full. 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