Recursive Constructions of Parallel FIFO and LIFO Queues with Switched Delay Lines

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1 Recursive Constructions of Parae FIFO and LIFO Queues with Switched Deay Lines Po-Kai Huang, Cheng-Shang Chang, Feow, IEEE, Jay Cheng, Member, IEEE, and Duan-Shin Lee, Senior Member, IEEE Abstract One of the most popuar approaches for the constructions of optica buffers needed for optica packet switching is to use Switched Deay Lines (SDL). Recent advances in the iterature have shown that there exist systematic SDL construction theories for various types of optica buffers, incuding First In First Out (FIFO) mutipexers, FIFO queues, priority queues, inear compressors, non-overtaking deay ines, and fexibe deay ines. As parae FIFO queues with a shared buffer are widey used in many switch architectures, e.g., input-buffered switches and oad-baanced Birkhoff-von Neumann switches, in this paper we propose a new SDL construction for such queues. The key idea of our construction for parae FIFO queues with a shared buffer is two-eve caching, where we construct a dua-port random request queue in the upper eve (as a high switching speed storage device) and a system of scaed parae FIFO queues with a shared buffer in the ower eve (as a ow switching speed storage device). By determining appropriate dumping threshods and retrieving threshods, we prove that the two-eve cache can be operated as a system of parae FIFO queues with a shared buffer. Moreover, such a two-eve construction can be recursivey expanded to an n-eve construction, where we show that the number of 2 2 switches needed to construct a system of N parae FIFO queues with a shared buffer B is O((N og N) og(b/(n og N))) for N >>. For the case with N =, i.e., a singe FIFO queue with buffer B, the number of 2 2 switches needed is O(og B). This is of the same order as that previousy obtained by Chang et a. We aso show that our two-eve recursive construction can be extended to construct a system of N parae Last In First Out (LIFO) queues with a shared buffer by using the same number of 2 2 switches, i.e., O((N og N) og(b/(n og N))) for N >> and O(og B) for N =. Finay, we show that a great advantage of our construction is its faut toerant capabiity. The reiabiity of our construction can be increased by simpy adding extra optica memory ces (the basic eements in our construction) in each eve so that our construction sti works even when some of the optica memory ces do not function propery. Index Terms Caches, FIFO queues, LIFO queues, optica buffers, switched deay ines. I. INTRODUCTION One of the key probems of optica packet switching is the ack of optica buffers. Unike eectronic packets, optica packets cannot be easiy stopped, stored, and forwarded. The This research was supported in part by the Nationa Science Counci, Taiwan, R.O.C., under Contract NSC E , Contract NSC E , Contract NSC E , and the Program for Promoting Academic Exceence of Universities NSC E PAE. The authors are with the Institute of Communications Engineering, Nationa Tsing Hua University, Hsinchu 3003, Taiwan, R.O.C. (e-mai: pkhuang@gibbs.ee.nthu.edu.tw; cschang@ee.nthu.edu.tw; ds@cs.nthu.edu.tw; jcheng@ee.nthu.edu.tw). ony known way to store optica packets is to direct them via a set of optica switches through a set of fiber deay ines so that optica packets come out at the right pace and at the right time. Such an approach, known as Switched Deay Line (SDL) construction, has received a ot of attention recenty (see e.g., [] [5] and the references therein). Eary SDL constructions for optica buffers, incuding the shared-memory switch in [] and CORD (contention resoution by deay ines) in [2][3], focused more on the feasibiity of such an approach. On the other hand, recent advances in SDL constructions have shown that there exist systematic methods for the constructions of various types of optica buffers, such as First In First Out (FIFO) mutipexers in [4] [9], FIFO queues in [0], priority queues in [][2], and inear compressors, non-overtaking deay ines, and fexibe deay ines in [3]. In this paper, we focus on the constructions of optica parae FIFO queues with a shared buffer as such queues are crucia in switch design. For instance, the virtua output queues in input-buffered switches (see e.g., [6][7]) and the centra buffers in oad-baanced Birkhoff-von Neumann switches (see e.g., [8][9]) can a be impemented by using parae FIFO queues with a shared buffer. One of the main contributions of this paper is to provide a two-eve recursive construction of parae FIFO queues with a shared buffer. The key idea of our two-eve construction is caching (see e.g., [20] [22]). The upper eve in our construction is a random request queue (see Definition 4 in Section II) that can be viewed as a high switching speed storage device, whie the ower eve in our construction is a system of scaed parae FIFO queues with a shared buffer that can be viewed as a ow switching speed storage device. By determining appropriate dumping threshods and retrieving threshods, we show that the two-eve cache can be operated as a system of parae FIFO queues with a shared buffer. Moreover, such a twoeve construction can be recursivey expanded to an n-eve construction, where we show that the number of 2 2 switches needed to construct a system of N parae FIFO queues with a shared buffer B is O((N og N)og(B/(N og N))) for N >>. For the case with N =, i.e., a singe FIFO queue with buffer B, the construction compexity (in term of the number of 2 2 switches) is O(og B). This is of the same order as that in [0]. To our surprise, the two-eve recursive construction can be extended to construct a system of N parae LIFO queues with a shared buffer. The ony modification of our architecture is to use a system of scaed parae LIFO queues with a shared buffer in the ower eve. Therefore, it can aso be recursivey

2 2 expanded to an n-eve construction and the number of 2 2 switches needed for the system remains the same. For the case with N =, i.e., a singe LIFO queue with buffer size B, the construction compexity is O(og B), which is better than O( B) as obtained in [][2] (we note that the designs in [][2] are more genera and work for priority queues). We aso note that one of the advantages of our construction is its faut toerant capabiity. By adding extra optica memory ces (the basic eements in our construction) in each eve, the reiabiity of our construction can be easiy increased in the sense that our construction sti works even after some of the optica memory ces are broken. The paper is organized as foows: In Section II, we introduce basic construction eements, incuding optica memory ces, FIFO queues, and random request queues. In Section III, we propose our two-eve recursive construction for N parae FIFO queues with a shared buffer, the associated operation rues, and the main theorem. We show that the two-eve recursive construction can be further expanded to an n-eve construction that has a much ower construction compexity in terms of the number of 2 2 switches. The extension to parae LIFO queues with a shared buffer is reported in Section IV. The paper is concuded in Section V. In the foowing, we provide a ist of notations used in the paper for easy reference. Q(t): the set of packets in N parae FIFO (resp. LIFO) queues at the end of the t th time sot Q (t): the set of packets in eve at the end of the t th time sot Q 2 (t): the set of packets in eve 2 at the end of the t th time sot Q,i (t): the set of packets in the i th queue in eve at the end of the t th time sot Q 2,i (t): the set of packets in the i th queue in eve 2 at the end of the t th time sot k: a scaing factor or a frame size F i (t): the set of packets in the i th front queue at the end of the t th time sot (see Definition 5) T i (t): the set of packets in the i th tai queue at the end of the t th time sot (see Definition 6) R T : Retrieving threshod R T = + k N D T : Dumping threshod D T = R T + k R(t): the set of queues that have packets in eve 2 at the end of the (t ) th time sot I i (p, t): the departure index of packet p in the i th queue at the end of the t th time sot A. Optica Memory Ces II. BASIC NETWORK ELEMENTS In our previous papers [0][3], we used optica memory ces as basic network eements for the constructions of various types of optica queues. As in the constructions in [0][3], we assume that packets are of the same size. Moreover, time is sotted and synchronized so that a packet can be transmitted within a time sot. An optica memory ce (see Figure ) is constructed by a 2 2 optica crossbar switch and a fiber deay ine with one time sot (unit) of deay. As iustrated in [0][3], we can set the 2 2 crossbar switch to the cross state to write an arriving packet to the optica memory ce. By so doing, the arriving packet can be directed to the fiber deay ine with one time sot of deay. Once the write operation is competed, we then set the crossbar switch to the bar state so that the packet directed into the fiber deay ine keeps recircuating through the fiber deay ine. To read out the information from the memory ce, we set the crossbar switch to the cross state so that the packet in the fiber deay ine can be routed to the output ink. Fig.. An optica memory ce: (a) writing information (b) recircuating information (c) reading information Network eements that are buit by optica crossbar switches and fiber deay ines are caed Switched Deay Line (SDL) eements in the iterature (see e.g., [] [3]). Ceary, an optica memory ce that is constructed by a 2 2 switch and a fiber deay ine with one unit of deay in Figure is an SDL eement. A scaed SDL eement is said to be with scaing factor k if the deay in every deay ine is k times of that in the origina (unscaed) SDL eement. For instance, if we scae the fiber ength from to 2 in Figure, then it is a scaed optica memory ce with scaing factor 2. As the ength is now increased to 2, the scaed optica member ce with scaing factor 2 can be used for storing two packets. In genera, each packet in a scaed SDL eement with scaing factor k can be individuay accessed as can be seen in our eary papers (see e.g., [7][9][0][2][3]). However, in the proposed recursive constructions of parae FIFO and LIFO queues in this paper we ony need to access the packets contiguousy as a bock of k packets. In other words, in this paper we group every k packets into a bock and view a scaed SDL eement with scaing factor k as an unscaed SDL eement for a bock of k packets. For instance, if we group every two packets into a bock, then a scaed optica memory ce with scaing factor 2 can be viewed as an unscaed optica memory ce for a bock of two packets. This is the key observation that we wi use in our construction of parae FIFO and LIFO queues in this paper. In the foowing, we extend the optica memory ce (with a singe input and a singe output) to a dua-port optica memory ce. Definition (Dua-port optica memory ces) A dua-port optica memory ce in Figure 2 is an optica memory ce with one additiona I/O port. It consists of a 3 3 switch and a fiber deay ine with one unit of deay. The 3 3 switch has the foowing three connection states: accessing state by the first I/O port in Figure 2(a), recircuating state in Figure 2(b), and accessing state by the second I/O port in Figure 2(c).

3 3 up one position. Specificay, a discrete-time FIFO queue is formaized in the foowing definition in [0]. Fig. 2. The three connection states of a dua-port optica memory ce: (a) accessing state by the first I/O port (b) recircuating state (c) accessing state by the second I/O port As an optica memory ce, a dua-port optica memory ce can be used for storing exacty one packet. Moreover, the stored packet can be accessed by either one of the two I/O ports. With the additiona I/O port, we note that a packet arriving at one input of an I/O port may be first stored in a dua-port optica memory ce and then routed to the output of another I/O port in a different time sot. In Figure 3, we show a simpe construction of a dua-port optica memory ce by adding a 2 2 switch in front of an optica memory ce and another 2 2 switch after the optica memory ce. It is easy to see that the recircuating state in Figure 2(b) can be reaized by setting a the 2 2 switches in Figure 3 to the bar state. For the accessing states in Figure 2(a) and (c), the 2 2 switch in the midde of Figure 3 is set to the cross state. If it is accessed by the first I/O port, then the other two 2 2 switches are set to the bar state. On the other hand, the other two 2 2 switches are set to the cross state if it is accessed by the second I/O port. This shows that a duaport optica memory ce can be constructed by three 2 2 switches. Ceary, the construction in Figure 3 can reaize a of the six possibe connection states for the 3 3 switch in a dua-port optica memory ce. However, we note that we ony need the three connection states described in Definition for the constructions of parae FIFO and LIFO queues in this paper. The construction compexity (in terms of the number of 2 2 switches) is of the same order as that of a simiar construction using a of the six possibe connection states, but the contro mechanism is much simper than using a of the six possibe connection states. Definition 2 (FIFO queues) A singe FIFO queue with buffer B is a network eement that has one input ink, one contro input, and two output inks. One output ink is for departing packets and the other is for ost packets. Then the FIFO queue with buffer B satisfies the foowing four properties: (P) Fow conservation: arriving packets from the input ink are either stored in the buffer or transmitted through one of the two output inks. (P2) Non-iding: if the contro input is enabed, then there is aways a departing packet if there are packets in the buffer or there is an arriving packet. (P3) Maximum buffer usage: if the contro input is not enabed, then an arriving packet is ost ony when buffer is fu. (P4) FIFO: packets depart in the FIFO order. The definition of a singe FIFO queue can be easiy extended to parae FIFO queues with a shared buffer as foows: Definition 3 (Parae FIFO queues with a shared buffer) A system of N parae FIFO queues with a shared buffer B is a network eement that has one input ink, N contro inputs, and two output inks (see Figure 4). As in Definition 2, one output ink is for departing packets and the other is for ost packets. Aso, each one of the N FIFO queues is associated with a contro input under the constraint that at most one of the N contro inputs is enabed at any time instant. Then the system of N parae FIFO queues with a shared buffer B satisfies (P), (P2), and (P4) in Definition 2 for each FIFO queue. However, as the buffer is shared by the N FIFO queues, the maximum buffer usage property needs to be modified as foows: (P3N) Maximum buffer usage: if there is no departing packet at time t, then an arriving packet at time t is ost ony when buffer is fu. contro inputs N input port output port oss port N Fig. 3. A simpe construction of a dua-port optica memory ce B. Parae FIFO Queues In a FIFO queue, a packet joins the tai of the queue when it arrives. If the buffer of a FIFO queue is finite, then an arriving packet to a fu queue is ost. When a packet departs from the head of a FIFO queue, every packet in the queue moves Fig. 4. The N parae FIFO queues Note that it is possibe that one of the N queues is enabed at time t and there is sti no departing packet at time t. This happens when the enabed queue is empty at time t.

4 4 The construction of a singe FIFO queue with buffer B has been studied in [0]. It is shown in [0] that there is a threestage recursive construction for a FIFO queue, and that a FIFO queue with buffer B can be constructed by using O(og B) 2 2 switches. However, using the construction of a singe FIFO queue in [0] for the construction of a system of N parae FIFO queues may not be efficient as each FIFO queue needs to be constructed with the same amount of buffer. In this paper, we wi propose a new two-eve recursive construction that aows the buffer to be shared among the N parae FIFO queues. C. Optica Random Request Queues (RRQs) In this section, we introduce the notion of a Random Request Queue (RRQ). In an RRQ, the departing packet, instead of the first one in a FIFO queue, coud be any packet in the queue (incuding the arriving one). As there is no particuar order for departures, the construction compexity of an RRQ is expected to be much higher than that of a FIFO queue. In the foowing, we provide the forma definition for an RRQ. Definition 4 (Random request queues) As indicated in Definition 2 for a FIFO queue, an RRQ with buffer B is a network eement that has one input ink, one contro input, and two output inks. One output ink is for departing packets and the other is for ost packets. Index the position in the buffer from, 2,..., B. An arriving packet can be paced in any one of the B positions as ong as it is not occupied (note that it is impicity assumed that there exists an interna contro for the pacing of an arriving packet). For an RRQ, the fow conservation property in (P) of Definition 2 and the maximum buffer usage property in (P3) of Definition 3 are sti satisfied. The non-iding property in (P2) of Definition 2 is not needed. Moreover, (P4) needs to be modified as foows: (P4R) Random request: the contro input in an RRQ has the set of states {0,, 2,..., B + }. When the state of the contro input is not zero, we say the contro input is enabed. If the state of the contro input is i for i =, 2,..., B, then the packet in the i th position of the queue (if there is one) is sent to the output ink. If the state of the contro input is B +, then the arriving packet (if there is one) is sent to the output ink. Fig. 5. A construction of an optica RRQ with buffer B Now we show in Figure 5 a way to construct an optica RRQ with buffer B by a concatenation of B optica memory ces. As discussed in the previous section, an optica memory ce can be used for storing one arriving optica packet. To see the random request property in (P4R), we index the B buffer positions (optica memory ces) from eft to right. Suppose B that the i th optica memory ce is empty, then an arriving packet can be written into the i th optica memory ce by setting the 2 2 optica crossbar switch of the i th optica memory ce to the cross state and the other 2 2 optica crossbar switches to the bar state. On the other hand, if the i th optica memory ce is occupied and the state of the contro input is i, where i =, 2,..., B, then the packet stored in the i th optica memory ce can be routed to the output by setting the 2 2 optica crossbar switch of the i th optica memory ce to the cross state and the other 2 2 optica crossbar switches to the bar state. If there is an arriving packet and the state of the contro input is B +, then the arriving packet can be sent to the output ink immediatey by setting a the 2 2 optica crossbar switches to the bar state. Note that it is possibe for a packet stored at the i th optica memory ce to depart from the RRQ whie an arriving packet is routed to the i th optica memory ce at the same time. The probem of the construction in Figure 5 is the maximum buffer usage property. If a the B optica memory ces are occupied and there is no departing packet, then an arriving packet shoud be routed to the oss port. For this reason, one needs to add a 2 switch in front of the construction in Figure 5 for admission contro. However, in the ater deveopment, we ony operate a the RRQs in such a way that there is no buffer overfow. As such, the 2 switch needed for the construction of an RRQ is omitted for carity. Instead of using optica memory ces with a singe I/O port, one can use dua-port optica memory ces in Figure 5. This resuts in a dua-port RRQ with buffer B in Figure 6. Note that we need two contro inputs for a dua-port RRQ, one contro input is for the random request from the first output ink and the other is for the random request from the second output ink. The dua-port RRQ in Figure 6 satisfies the fow conservation property and the random request property in Definition 4. However, as there are ony three connection patterns in every dua-port optica memory ce, it is not possibe for an arriving packet to be routed to the i th dua-port optica memory ce from the input ink of one I/O port whie the packet stored in the i th dua-port optica memory ce is departing from the output ink of another I/O port at the same time. One consequence of such a restriction is that the maximum buffer usage property is not satisfied. This can be seen from the foowing worst-case scenario. Suppose that a of the B duaport optica memory ces are occupied at time t. If at time t+ there is a packet arriving at the input ink of the second I/O port and the state of the first contro input is i, then the packet stored in the i th dua-port optica memory ce wi be sent to the output ink of the first I/O port, but the arriving packet at the input ink of the second I/O port can not be paced in the i th dua-port optica memory ce (which is empty now). As such, the arriving packet at the second input ink has to be sent to the oss ink and the maximum buffer usage property is not satisfied. It is cear that if the maximum buffer usage property has to be satisfied, then the maximum buffer that can be achieved by the construction in Figure 6 is B. Therefore, it woud be technicay correct to ca the construction in Figure 6 a dua-port RRQ with buffer B. However, in the recursive

5 5 constructions of parae FIFO and LIFO queues in this paper, we never require an arriving packet from the input ink of one I/O port be routed to the output ink of another I/O port of a dua-port optica memory ce at the same time, and hence the worst-case scenario mentioned above never occurs. In other words, there are aways empty dua-port optica memory ces for arriving packets at the input inks (see the proof of Theorem 8 in Appendix A for detais). As such, in our proposed scheme the construction in Figure 6 achieves the maximum buffer B, and that is why we ca the construction in Figure 6 a dua-port RRQ with buffer B in this paper. Finay, we woud ike to point out that the reason why the maximum buffer usage property is not satisfied is due to the fact that we ony use three connection patterns in every duaport optica memory ce in this paper. If we use a of the six possibe connection states for the 3 3 switch in a duaport optica memory ce, then the maximum buffer usage property is satisfied. However, the maximum buffer that can be achieved is sti the same. This impies that a construction simiar to that proposed in this paper and using a of the six possibe connection states achieves the same order of buffer size, but undoubtedy increases the compexity of the contro mechanism. Fig. 6. A construction of a dua-port RRQ with buffer B via a concatenation of dua-port optica memory ces III. RECURSIVE CONSTRUCTIONS OF PARALLEL FIFO QUEUES WITH A SHARED BUFFER A. A Two-eve Construction of Parae FIFO Queues with a Shared Buffer B It is obvious to see that an RRQ with buffer B can be operated as N parae FIFO queues with a shared buffer B. However, the number of 2 2 switches needed for the construction of an RRQ with buffer B in Figure 5 is aso B. As packets have to depart in the FIFO order, the construction compexity of N parae FIFO queues with a shared buffer B (in terms of the number of 2 2 switches) shoud be much ess than that of an RRQ with buffer B. To show this, in this section we provide a recursive construction of N parae FIFO queues with a shared buffer B + kb 2 in Figure 7. The construction in Figure 7 consists of two eves: a dua-port RRQ with buffer B in eve, and a scaed SDL network eement that can be used as a system of N parae FIFO queues with a shared buffer B 2 and scaing factor k in eve 2. The 2 switch in front of the network eement is for admission contro. Its objective is to make sure that the tota number of packets inside the network eement does not exceed B +kb 2. An arriving packet can ony be admitted if the tota number of packets inside the network eement does not exceed B + kb 2 after its admission. Otherwise, it is routed to the oss port. input port oss port eve Dua-port optica RRQ with buffer B eve 2 N parae FIFO queues with buffer B 2 and scaing factor k output port Fig. 7. A recursive construction of N parae FIFO queues with buffer B + kb 2 The key idea behind the construction in Figure 7 is caching. Note that if we group every k time sots into a frame and operate the scaed SDL eement in eve 2 at the time scae of frames, then the scaed SDL eement in eve 2 can be used as a system of N parae FIFO queues with a shared buffer B 2 by viewing k consecutive packets as a bock of packets. As such, the scaed SDL eement in eve 2 can be viewed as a storage device with a much ower switching speed (k times sower) than that of the dua-port RRQ in eve. As in most caching systems, the probems are about (i) when to dump packets from the high switching speed storage device in eve to the ow switching speed storage device in eve 2, and (ii) when to retrieve packets from the ow switching speed storage device in eve 2 to the high switching speed storage device in eve. Consequenty, we et D T be the dumping threshod and R T be the retrieving threshod. These two threshods wi be used to determine when to dump packets and when to retrieve packets. To be precise, et Q,i (t), = and 2, i =, 2,...,N, be the set of packets in the i th queue that are stored in eve at the end of the t th time sot. Then the set of packets in the i th queue at the end of the t th time sot is simpy Q,i (t) Q 2,i (t). Furthermore, et Q (t) (resp. Q 2 (t)) be the set of packets in eve (resp. eve 2) at the end of the t th time sot. Ceary, for = and 2, Q (t) = N Q,i (t). () i= Aso, the set of packets in the N parae FIFO queues at the end of the t th time sot, denoted by Q(t), is the union of the set of packets in each queue of each eve, i.e., Q(t) = Q (t) Q 2 (t) = 2 i= N Q,i (t). (2) For a the packets in the i th FIFO queue at time t, i.e., Q,i (t) Q 2,i (t), we can sort them according to their departure order. Specificay, we et I i (p, t) be the departure index of packet p in the i th queue at time t, i.e., I i (p, t) = j if packet p is the j th packet to depart in the i th queue at the end of the t th time sot. In the foowing, we use the departure index to define the notions of front queues and tai queues that are needed for our operation.

6 6 Definition 5 (Front queues) The i th front queue at time t, denoted by F i (t), is a subset of the packets in the i th queue in eve at time t, i.e., F i (t) Q,i (t). A packet p is in F i (t) if () there are packets in the i th queue in eve 2 and the departure index of packet p is smaer than that of any packet in the i th queue in eve 2, i.e., Q 2,i (t) > 0 and I i (p, t) < I i ( p, t), p Q 2,i (t), or (2) there are no packets in the i th queue in eve 2 and the departure index of packet p is not greater than the dumping threshod, i.e., Q 2,i (t) = 0 and I i (p, t) D T. Definition 6 (Tai queues) The i th tai queue at time t, denoted by T i (t), is a subset of the packets in the i th queue in eve at time t, i.e., T i (t) Q,i (t). A packet p is in T i (t) if () there are packets in the i th queue in eve 2 and the departure index of packet p is greater than that of any packet in the i th queue in eve 2, i.e., Q 2,i (t) > 0 and I i (p, t) > I i ( p, t), p Q 2,i (t), or (2) there are no packets in the i th queue in eve 2 and the departure index of packet p is greater than the dumping threshod, i.e., Q 2,i (t) = 0 and I i (p, t) > D T. We note from Definition 5 and Definition 6 that the departure index of a packet in the i th front queue is aways smaer than that of any packet in the i th tai queue at any time, no matter the i th queue in eve 2 is empty or not. As such, the i th front queue and the i th tai queue are aways disjoint at any time, i.e., F i (t) T i (t) = φ for a t. Now we describe the operations of our recursive construction in Figure 7. In our operations, every k time sots are grouped into a frame. The RRQ in eve is operated in every time sot, whie the scaed N parae FIFO queues in eve 2 is operated in the time scae of frames. (R0) Admission contro: an arriving packet can be admitted to the network eement in Figure 7 ony if the tota number of packets in the network eement does not exceed B +kb 2 after its admission. Otherwise, it is routed to the oss port by the 2 switch in front of the network eement in Figure 7. (R) Write operation: suppose that there is an arriving packet to the i th queue at time t. If the i th queue is empty at time t and the i th queue is enabed at time t, then the arriving packet is routed to the output port immediatey. Otherwise, the arriving packet is stored in the dua-port RRQ in eve (as ong as the tota number of packets in the construction does not exceed B + kb 2 after its admission). (R2) Read operation: suppose that the i th queue is enabed at time t. If the i th queue is empty at time t and there is an arriving packet to the i th queue at time t, then the arriving packet is routed to the output port immediatey. If the i th queue has packets in eve at time t, the packet that has the smaest departure index among a the packets of the i th queue in the dua-port RRQ in eve is sent to the output port at time t. Otherwise, there is no departing packet at time t. (R3) Retrieve operation (the shortest front queue beow the retrieving threshod): suppose that t is the beginning time sot of the m th frame, i.e., t = k(m ) +. Consider the set of queues R(t) that have packets in eve 2 at time t. Suppose that the i th queue is the queue that has the smaest number of packets in its front queue at time t among a the queues in R(t). If the number of packets in the i th front queue at time t is ess than or equa to the retrieving threshod R T, i.e., F i (t ) R T, then the i th FIFO queue in eve 2 is enabed during the m th frame. As such, there are k packets retrieved from the i th FIFO queue in eve 2 to the i th front queue in [t, t + k ]. (R4) Dump operation (the ongest tai queue with a fu bock of packets): suppose that t is the beginning time sot of the m th frame, i.e., t = k(m ) +. Suppose that the i th queue is the queue that has the argest number of packets in its tai queue at time t among a the N queues. If there are at east k packets in the i th tai queue at time t, i.e., T i (t ) k, then the k packets with the smaest departure indices in the i th tai queue are sent (starting from the packet with the smaest departure index among these k packets) to the i th FIFO queue in eve 2 (as a bock of packets) provided that there is buffer space in eve 2 (i.e., either the buffer of the N FIFO queues in eve 2 is not fu at time t or there is a retrieve operation at time t). We note that both the retrieve operation and the dump operation can ony occur at the beginning time sot of a frame. Aso, if the two-eve recursive construction in Figure 7 is started from an empty system, then in our operations we aways keep Q,i (t) = F i (t) T i (t). In other words, if the i th queue in eve 2 is not empty, then the departure index of a packet in the i th queue in eve is either greater than that of any packet of the i th queue in eve 2 or smaer than that of any packet of the i th queue in eve 2. (As for the case that the i th queue in eve 2 is empty, it hods triviay that Q,i (t) = F i (t) T i (t) as in this case F i (t) contains a the packets in the i th queue with departure indices ess than or equa to the dumping threshod D T, and T i (t) contains a the packets in the i th queue with departure indices greater than D T.) As this property wi be very usefu in the proof of our main theorem (Theorem 8 beow) in this paper, we state this property formay in the foowing emma. Lemma 7 Suppose that the two-eve recursive construction in Figure 7 is started from an empty system. Then under (R0) (R4), we have Q,i (t) = F i (t) T i (t) for a t 0. Proof. We prove this emma by induction on t. As the twoeve recursive construction in Figure 7 is started from an empty system, Q,i (t) = F i (t) T i (t) hods triviay for t = 0.

7 7 Assume that it aso hods for some t 0. We consider the foowing four cases. Case : There is an arriving packet to the i th queue at time t. According the write operation in (R), the arriving packet is either routed to the output port immediatey or stored in the dua-port RRQ in eve (as ong as the tota number of packets in the construction does not exceed B + kb 2 after its admission). If the arriving packet is routed to the output port immediatey or the i th queue in eve 2 is empty at time t, then there is nothing to prove. On the other hand, if the arriving packet is not routed to the output port immediatey and the i th queue in eve 2 is not empty at time t, then it wi be paced in the i th tai queue as the arriving packet has the argest departure index among a the packets of the i th queue. From the induction hypothesis, we easiy concude that Q,i (t) = F i (t) T i (t). Case 2: The i th queue is enabed at time t. According the read operation in (R2), the packet with the smaest departure index among a the packets, incuding the arriving packet (if there is one), of the i th queue in the dua-port RRQ in eve is sent to the output port at time t. It foows triviay from the induction hypothesis that Q,i (t) = F i (t) T i (t). Case 3: There is a retrieve operation performed on the i th queue at time t. According the retrieve operation in (R3), the retrieved packet is the packet with the smaest departure index among a the packets in the i th FIFO queue in eve 2 as packets in a FIFO queue must depart in the FIFO order. If the i th queue in eve 2 is empty at time t, then there is nothing to prove. On the other hand, if the i th queue in eve 2 is not empty at time t, then the retrieved packet wi be paced in the i th front queue as it has a departure index smaer than a the packets of the i th queue in eve 2 at time t. As such, we have from the induction hypothesis that Q,i (t) = F i (t) T i (t). Case 4: There is a dump operation performed on the i th queue at time t. According the dump operation in (R4), the dumped packet is the packet with the smaest departure index among a the packets of the i th tai queue. As in Case 2 above, it foows triviay from the induction hypothesis that Q,i (t) = F i (t) T i (t). Finay, we note that athough we discuss the above four cases separatey, the arguments sti hod if two or more of the above four cases occur at the same time. Now we state the main theorem of our paper. The proof of Theorem 8 wi be presented in Appendix A. Theorem 8 Suppose the two-eve recursive construction in Figure 7 is started from an empty system. If we choose R T = + k N, D T = R T + k, and B ND T + N(k )+k+, then under (R0) (R4) the construction in Figure 7 achieves the exact emuation of a system of N parae FIFO queues with a shared buffer B + B 2 k. To see the intuition why we need to set R T =. We consider an idea fuid mode as in [2] + k N and [22]. As can be seen in [2] and [22], the argest amount of fuid that can be drained from queue in eve is achieved in the foowing scenario. Suppose that initiay the number of packets in every front queue in eve is R T +ǫ for some sma ǫ > 0. As such, no retrieve operation is performed. During the first frame, a the N front queues in eve are drained at the same rate. By the end of the first frame, the number of packets in the i th front queue, i =, 2,..., N, is roughy R T +ǫ k/n. At the beginning of the second frame, a retrieve operation is performed on one of the N queues, say queue N. This takes another k time sots (a frame) and k packets of queue N are retrieved from eve 2 to its front queue in eve. During the second frame, the first N front queues are drained at the same rate. By the end of the second frame, the number of packets in the i th front queue, i =, 2,..., N, is roughy R T +ǫ k/n k/(n ). At the beginning time sot of the third frame, a retrieve operation is performed on queue N. During the third frame, the first N 2 front queues are drained at the same rate. By the end of the third frame, the number of packets in the i th front queue, i =, 2,..., N 2, is roughy R T +ǫ k/n k/(n ) k/(n 2). Repeating the same argument, one can argue that by the end of the N th frame the number of packets in the first front queue is roughy R T + ǫ k N /. This has to be nonnegative so that the non-iding property can be satisfied. We now discuss our choice of D T, the threshod for dump operations. If we set D T R T + k, then from the definitions of a front queue and a tai queue, a the k packets retrieved in one frame from a queue in eve 2 wi be stored in its front queue in eve. Since a arger D T woud require a arger buffer size B for the dua-port RRQ in eve, we set D T = R T + k. The reason why we need B ND T + N(k ) + k + can be expained intuitivey by the foowing scenario: suppose at the beginning time sot of a frame each of the N queues has D T packets in its front queue and k packets in its tai queue. As such, no dump operation is performed for that frame. During that frame, there are k arriving packets and they are stored in the dua-port RRQ in eve. At the end of that frame, there are N(D T + k ) + k packets in the dua-port RRQ in eve. As such, one of the tai queues must have at east k packets and a dump operation is performed at the beginning time sot of the next frame. Suppose that there is another arriving packet at the beginning time sot of the next frame. Even though there is a packet dumped from eve to eve 2, this arriving packet has to be stored in a buffer space different from the one being freed by the dumped packet. This is because the dua-port RRQ in eve, constructed by dua-port optica memory ces, ony aows three connection patterns and the dumped packet and the arriving packet have to use different I/O ports. As such, we need B ND T + N(k )+k + at the beginning time sot of the next frame in this scenario. Furthermore, if in the above scenario there is an arriving packet in each time sot, then the dump operations continue unti the buffer in eve 2 is fu, from which time the arriving packets are stored in the buffer in eve unti the entire queue is fu. This shows that the maximum possibe buffer B +B 2 k coud be achieved.

8 8 In short, our choices of B and D T ensure that empty memory ces are aways avaiabe to store new arriving packets. As such, the fow conservation property and the maximum buffer usage property are satisfied. Aso, our choice of R T ensures that the non-iding property is satisfied. B. Recursive Expansion to an n-eve Construction of Parae FIFO Queues with a Shared Buffer One can recursivey expand the two-eve construction in Theorem 8 to an n-eve construction in Figure 8. This n-eve construction can be used for exact emuation of a system of N parae FIFO queues with a shared buffer B (k n )/(k ) + B 2 k n. To see this, consider the case when n = 3. Then we have from Theorem 8 that the dua-port RRQ with buffer B and scaing factor k in eve 2 and the system of N parae FIFO queues with a shared buffer B 2 and scaing factor k 2 in eve 3 can be used for exact emuation of a system of N parae FIFO queues with a shared buffer B + kb 2 and scaing factor k. Using Theorem 8 again, one can show that the 3-eve construction can be used for exact emuation of a system of N parae FIFO queues with a shared buffer B + k(b + kb 2 ). Fig. 8. input port eve Dua-port optica RRQ with buffer size B eve2 Dua-port optica RRQ with buffer size B and scaing factor k eve n- Dua-port optica RRQ with buffer size B and scaing factor k n-2 eve n N parae FIFO queues with buffer size B 2 and scaing factor k n- An n-eve construction of N parae FIFO queues output port Note that an RRQ with buffer B can be used for exact emuation of a system of N parae FIFO queues with a shared buffer B and that the number of 2 2 switches needed for an RRQ with buffer B in Figure 5 is O(B). If we choose B 2 = B in Figure 8, then the number of 2 2 switches needed for the n-eve construction of N parae FIFO queues with buffer B (k n )/(k ) is O(nB ). This shows that one can construct N parae FIFO queues with buffer B with O(B og k (B/B )) 2 2 switches. From Theorem 8, the minimum number that one can choose for B is ND T +N(k )+k+. For N =, one can simpy choose B = 4k+ and construct a FIFO queue with O(og B) 2 2 switches. This is of the same order as that in [0]. In particuar, if we choose k = 2, then one ony needs 9 dua-port optica memory ces in each eve. In Figure 9, we show a 3-eve construction of a FIFO queue with buffer Fig. 9. A 3-eve construction of a FIFO queue with buffer 63 For N >> and k = 2, we know from the compexity of the harmonic function that B = ND T + N(k ) + k + is O(N og N). Thus, O((N og N)og(B/(N og N))) 2 2 switches can be used to construct N parae FIFO queues with buffer B. Note that in Theorem 8 the condition for the buffer in eve is B ND T + N(k ) + k +. This eads to a great advantage of our construction the faut toerant capabiity. Specificay, if each optica memory ce has a bypass circuit that sets up a direct connection between its input ink and its output ink once a faut within the optica memory ce is detected, then by setting B = F +ND T +N(k )+k + our construction sti works even after up to F optica memory ces are broken. In [23], Bouiard and Chang provided a soution for the contro of 2 2 switches in optica FIFO queues and nonovertaking deay ines, which were designed based on recursive constructions. As in this paper the parae FIFO queues with a shared buffer are aso recursivey constructed, and from (R0) (R4) we know that we ony need to keep track of the front queues and tai queues of the RRQs, the contro mechanism of the 2 2 switches in the proposed recursive construction of parae FIFO queues with a shared buffer is expected to be simper than that in [23]. Before we move on to the recursive constructions of parae LIFO queues with a shared buffer in the next section, we briefy address a few practica impementation issues that are of concern to some researchers. With recent advances in optica technoogies, the constructions of compact and tunabe optica buffers have been made feasibe by using the so-caed sow ight technique [24] [28]. For instance, optica buffers can be impemented in the nano-scae in today s technoogy [24], and hence the construction of an optica buffer may not be as buky as one might expect. Aso it has been demonstrated that a 75-ps puse can be deayed by up to 47 ps [28], and thus the synchronization issue that is usuay of practica concern may not be a serious design obstace. Furthermore, current photonic technoogy aows to impement a 2 2 optica memory ce using photonic regeneration and reshaping (P2R) waveength converters [29] [3] and arrayed waveguide grating (AWG) [32][33]. According to [3], P2R waveength converters have an exceent cascadabiity of up to fifteen cascaded stages using today s technoogy. With error correcting codes empoyed, it is expected that the number of cascaded stages can be much higher. As such, the power

9 9 oss due to recircuations through the fiber deay ines may not be a critica design obstace as one might expect. Of course, it wi be unreaistic to aow a packet to recircuate through the fiber deay ines indefinitey, and there shoud be a imitation on the number of recircuations through the fiber deay ines. In an approximate impementation, one may simpy drop the packets that have to be recircuated through the fiber deay ines for more than a certain number of times. Aternativey, one may take into consideration the imitation on the number of recircuations through the fiber deay ines during the design process. We have made some progresses on the constructions of optica 2-to- FIFO mutipexers with a imited number of recircuations, and the resuts wi be reported ater in a separate paper. Finay, we mention that crosstak interference is aso a practica impementation issue of concern. We have made some progresses on the constructions of inear compressors with minimum crosstak, and resuts aong this ine wi be reported ater in a separate paper. IV. RECURSIVE CONSTRUCTIONS OF PARALLEL LIFO QUEUES WITH A SHARED BUFFER We have proposed a recursive construction in Figure 7 to construct parae FIFO queues. One key condition that makes such a construction feasibe is the constraint of FIFO order among the arriving packets. In this section, we wi show that parae LIFO queues can aso be constructed using a simiar architecture. input port oss port eve Dua-port optica RRQ with buffer B eve 2 N parae LIFO queues with buffer B 2 and scaing factor k output port Fig. 0. A recursive construction of N parae LIFO queues with buffer B + kb 2 The definition of N parae LIFO queues is the same as that for N parae FIFO queues except that packets depart in the Last In First Out (LIFO) order. In Figure 0, the construction consists of two eves: a dua-port RRQ with buffer B in eve, and a scaed SDL network eement that can be used as a system of N parae LIFO queues with a shared buffer B 2 and scaing factor k in eve 2. The 2 switch in front of the network eement is for admission contro. Its objective is to make sure that the tota number of packets inside the network eement does not exceed B + kb 2. An arriving packet can ony be admitted if the tota number of packets inside the network eement does not exceed B +kb 2 after its admission. Otherwise, it is routed to the oss port. We use the same notations as we did in the construction of parae FIFO queues. Specificay, we et Q,i (t), = and 2, i =, 2,, N, be the set of packets in the i th queue that are stored in eve at the end of the t th time sot, and et I i (p, t) be the departure index of packet p at time t, i.e., I i (p, t) = j if packet p is the j th packet to depart in the i th queue at the end of the t th time sot. Here, the departure index is abeed according to LIFO order. Moreover, as the operations for N parae FIFO queues, the RRQ in eve is operated in every time sot, whie the scaed N parae LIFO queues in eve 2 is operated in the time scae of frames. Note that the notations of front queues and tai queues are no onger needed, because under our operation rues the departure index of a packet stored in eve is aways ower than that of any packet stored in eve 2. Now we present the operation rues of the recursive construction in Figure 0. The rue for admission contro is the same as that in (R0). However, the write operation rue, the read operation rue, the retrieve operation rue, and the dump operation rue need to be modified as foows: (LR) Write operation: suppose that there is an arriving packet to the i th queue at time t. If the i th queue is enabed at time t, then the arriving packet is routed to the output port immediatey. Otherwise, the arriving packet is stored in the RRQ in eve (as ong as the tota number of packets in the construction does not exceed B + kb 2 after its admission). (LR2) Read operation: suppose that the i th queue is enabed at time t. If there is an arriving packet to the i th queue at time t, then the arriving packet is routed to the output port immediatey. If there is no arriving packet to the i th queue at time t and the i th queue has packets in eve at time t, the packet that has the smaest departure index among a the packets of the i th queue in the RRQ in eve is sent to the output port at time t. Otherwise, there is no departing packet at time t. (LR3) Retrieve operation (the shortest queue beow the retrieving threshod): suppose that t is the beginning time sot of the m th frame, i.e., t = k(m ) +. Consider the set of queues R(t) that have packets in eve 2 at time t. Suppose that the i th queue is the queue that has the smaest number of packets at time t among a the queues in R(t). If the number of packets in the i th queue at time t is ess than or equa to the retrieving threshod R T, i.e., Q,i (t ) R T, then the i th LIFO queue in eve 2 is enabed during the m th frame. As such, there are k packets retrieved from the i th LIFO queue in eve 2 to the i th queue in eve in [t, t + k ]. (LR4) Dump operation (the ongest queue with a fu bock of packets): suppose that t is the beginning time sot of the m th frame, i.e., t = k(m )+. Suppose that the i th queue is the queue that has the argest number of packets at time t among a the N queues. If there are at east D T +k packets in the i th queue in eve at time t, i.e., Q,i (t ) D T + k, then the k packets with the argest departure indices in the i th queue in eve are sent (starting from the packet with the smaest departure index among

10 0 these k packets) to the i th LIFO queue in eve 2 (as a bock of packets) provided that there is buffer space in eve 2 (i.e., either the buffer of the N LIFO queues in eve 2 is not fu at time t or there is a retrieve operation at time t). Now we state the main resut for the construction of parae LIFO queues in the foowing theorem. The proof of Theorem 9 is given in Appendix B. Theorem 9 Suppose the two-eve recursive construction in Figure 0 is started from an empty system. If we choose R T = + k N, D T = R T + k, and B ND T + N(k )+k+, then under (R0) and (LR) (LR4) the construction in Figure 0 achieves the exact emuation of a system of N parae LIFO queues with a shared buffer B + B 2 k. The intuition for the choice of R T and B is the same as that in section Section III. Moreover, we can aso expand the two-eve construction in Theorem 9 to an n-eve construction in Figure. Fig.. input port eve Dua-port optica RRQ with buffer size B eve2 Dua-port optica RRQ with buffer size B and scaing factor k eve n- Dua-port optica RRQ with buffer size B and scaing factor k n-2 eve n N parae LIFO queues with buffer size B 2 and scaing factor k n- An n-eve construction of N parae LIFO queues output port Since we use the same construction for parae FIFO queues and parae LIFO queues, the number of 2 2 switches needed for the two systems are the same. For a singe LIFO queue with buffer size B (the case with N = ), the construction compexity is O(og B), which is better than O( B) as given in [][2] (we note that the constructions in [][2] are more genera and work for priority queues). Moreover, as we do not need to distinguish between the front queue and the tai queue as in the construction for parae FIFO queues, the contro of the parae LIFO queues is much easier than that for the parae FIFO queues. V. CONCLUSIONS In this paper, we provide a new two-eve recursive construction of a system of parae FIFO (resp. LIFO) queues with a shared buffer. The key idea of our two-eve construction is caching, where we have a dua-port RRQ in eve that acts as a high switching speed storage device and a system of scaed parae FIFO (resp. LIFO) queues in eve 2 that acts as a ow switching speed storage device. By determining appropriate dumping threshods and retrieving threshods, we prove that the two-eve cache can indeed be operated as a system of parae FIFO (resp. LIFO) queues with a shared buffer. We have shown that one of the advantages of our construction is its faut toerant capabiity. By adding extra optica memory ces in each eve, our construction sti works even after some of the optica memory ces are broken. Furthermore, to construct a singe LIFO queue with buffer size B, our construction ony needs O(og B) 2 2 switches, which is sharper than O( B) previousy given in [][2] (we note that the constructions in [][2] are more genera and work for priority queues). There are some extensions that need to be further expored. (i) N-port optica memory ces: for this paper, a dua-port optica RRQ in Figure 6 is constructed. Using the same architecture, an N-port optica RRQ can be constructed via a concatenation of N-port optica memory ces. It woud be of interest to ook for the efficient construction of an N-port optica memory ce. (ii) N-to- mutipexer: the key condition to make our twoeve recursive constrution feasibe is the constraint of FIFO order or LIFO order among the arriving packets. Since an N-to- mutipexer has a simiar constraint, is it possibe to do the recursive construction of an N-to- mutipexer with an (N + )-port optica RRQ in eve and a scaed N-to- mutipexer in eve 2? (iii) N N output-buffered switch: based on the same reason of (ii), is it possibe to do the recursive construction of an N N output-buffered switch with a 2N-port optica RRQ in eve and a scaed N N output-buffered switch in eve 2? APPENDIX A PROOF OF THEOREM 8 In this appendix, we prove Theorem 8. The proof of Theorem 8 is based on the foowing three emmas that bound the size of front queues and tai queues. In Lemma 0, we first show upper bounds for tai queues. We then show upper bounds for front queues in Lemma and ower bounds for front queues in Lemma 2. Lemma 0 Suppose that t is the beginning time sot of a frame. (i) Suppose that no dump operation is performed at time t. If either the buffer of the N parae FIFO queues in eve 2 is not fu at time t or there is a retrieve operation at time t, then N T j (t ) N(k ). (ii) If the buffer of the N parae FIFO queues in eve 2 is not fu at time t, then N T j (t ) N(k ) + k.