Ching-Ming Lien

Load balanced Birkhoff-von Neumann switches, part II: multi-stage buffering

The main objective of this sequel is to solve the out-of-sequence problem that occurs in the load balanced Birkhoff-von Neumann switch with one-stage buffering. We do this by adding a load-balancing buffer in front of the first stage and a resequencing-and-output buffer after the second stage. Moreover, packets are distributed at the first stage according to their flows, instead of their arrival times in part I. In this paper, we consider multicasting flows with two types of scheduling policies: the first come first served (FCFS) policy and the earliest deadline first (EDF) policy. The FCFS policy requires a jitter control mechanism in front of the second stage to ensure proper ordering of the traffic entering the second stage. For the EDF scheme, there is no need for jitter control. It uses the departure times of the corresponding FCFS output-buffered switch as deadlines and schedules packets according to their deadlines. For both policies, we show that the end-to-end delay through our multi-stage switch is bounded above by the sum of the delay from the corresponding FCFS output-buffered switch and a constant that only depends on the size of the switch and the number of multicasting flows supported by the switch.

On Constructions of Optical Queues with a Limited Number of Recirculations

Recently, there has been a lot of attention on the constructions of optical queues by using optical Switches and fiber Delay Lines (SDL). In this paper, we consider the constructions of optical queues with a limited number of recirculations through the fibers in such SDL constructions. Such a limitation on the number of recirculations comes from practical feasibility considerations, such as crosstalk, power loss, amplified spontaneous emission (ASE) from the Erbium doped fiber amplifiers (EDFA), and the pattern effect of the optical switches. We first transform the design of the fiber delays in such SDL constructions to an equivalent integer representation problem. Specifically, given 1 les k les M, we seek for an M-sequence dM 1 = (d1,d2,...,dm) of positive integers to maximize the number of consecutive integers (starting from 0) that can be represented by the C-transform relative to dM 1 such that there are at most k 1-entries in their C-transforms. Then we give a class of greedy constructions so that d1, d2,..., dM are obtained recursively and the maximum number of representable consecutive integers by using d1,d2,...,di is larger than that by using d1,d2,...,di-1 for all i. Furthermore, we obtain an explicit recursive expression for d1, d2,..., dM given by a greedy construction. Finally, we show that an optimal M-sequence (in the sense of achieving the maximum number of representable consecutive integers) can be given by a greedy construction. The solution of such an integer representation problem can be applied to the construction of optical 2-to-l FIFO multiplexers with a limited number of recirculations. We show that the complexity of searching for an optimal construction under our routing policy can be greatly reduced from exponential time to polynomial time by only considering the greedy constructions instead of performing an exhaustive search. Similar results can be obtained for linear compressors and linear decompressors with a limited number of recirculations.

Load balanced Birkhoff-von Neumann switches with resequencing

In [2], we proposed the load balanced Birkhoff-von Neumannswitch with one-stage buffering (see Figure 1). Such a switchconsists of two stages of crossbar switching fabrics and one stageof buffering. The buffer at the input port of the second stage usesthe Virtual Output Queueing (VOQ) technique to solve the problem ofhead-of-line blocking. In such a switch, packets are of the samesize. Also, time is slotted and synchronized so that exactly onepacket can be transmitted within a time slot. In a time slot, bothcrossbar switches set up connection patterns corresponding topermutation matrices that are periodically generated from aone-cycle permutation matrix. The reasoning behind such a switch architecture is as follows:since the connection patterns are periodic, packets from the sameinput port of the first stage are distributed in a round-robinfashion to the second stage according to their arrival times. Thus,the first stage performs load balancing for the incoming traffic.As the traffic coming into the second stage is load balanced, itsuffices to use simple periodic connection patterns to performswitching at the second stage. This is shown in [2] as a specialcase of the original Birkhoff-von Neumann decomposition used in[1]. There are several advantages of using such an architecture,including scalability, low hardware complexity, 100% throughput,low average delay in heavy load and bursty traffic, and efficientbuffer usage. However, the main drawback of the load balancedBirkhoff-von Neumann switch with one-stage buffering is thatpackets might be out of sequence. The main objective of this paper is to solve the out-of-sequenceproblem that occurs in the load balanced Birkhoff-von Neumannswitch with one-stage buffering. One quick fix is to add aresequencing-and-output buffer after the second stage. However, aspackets are distributed according to their arrival times atthe first stage, there is no guarantee on the size of theresequencing-and-output buffer to prevent packet losses. For this,one needs to distributed packets according to their flows,as indicated in the paper by Iyer and McKeown [5]. This is done byadding a flow splitter and a load-balancing buffer in front of thefirst stage (see Figure 2). For an N x N switch, theload-balancing buffer at each input port of the first stageconsists of N virtual output queues (VOQ) destined for theN output ports of that stage. Packets form the sameflow are split in the round-robin fashion to the Nvirtual output queues and scheduled under the First Come FirstServed (FCFS) policy. By so doing, load balancing can be achievedfor each flow as packets from the same flow are split almost evenlyto the input ports of the second stage. More importantly, aspointed out in [5], the delay and the buffer size of theload-balancing buffer are bounded by constants that only depend onthe size of the switch and the number of flows. Theresequencing-and-output buffer after the second stage not onlyperforms resequencing to keep packets in sequence, but also storespackets waiting for transmission from the output links. In this paper, we consider a traffic model with multicastingflows. This is a more general model than the point-to-point trafficmodel in [5]. A multicasting flow is stream of packets that has onecommon input and a set of common outputs. For the multicastingflows, fanout splitting (see e.g., [4]) is performed at the centralbuffers (the VOQ in front of the second stage). The central buffersare assumed to be infinite so that no packets are lost in theswitch. We consider two types of scheduling policies in the centralbuffers: the FCFS policy and the Earliest Deadline First (EDF)policy. For the FCFS policy, a jitter control mechanism, is addedin the VOQ in front of the second stage. Such a jitter controlmechanism delays every packet to its maximum delay at the firststage so that the flows entering the second stage are simplytime-shifted flows of the original ones. Our main result for theFCFS scheme with jitter controls is the following theorem. Theproof of Theorem 1 is shown in the full report [3]. Theorem 1 Suppose that all the buffers are empty attime 0. Then the followings hold for FCFS scheme with jittercontrol. (i) The end-to-end delay for a packet through our switch withmulti-stage buffering is bounded above by the sum of the delaythrough the corresponding FCFS output-buffered switch and NLmax + (N + 1)Mmax,where Lmax (resp. Mmax) isthe maximum number of flows at an input (resp. output)port. (ii) The load-balancing buffer at an input port of the firststage is bounded above by N Lmax. (iii) The delay through the load-balancing buffer at an inputport of the first stage is bounded above by NLmax. (iv) The resequencing-and-output buffer at an output port ofthe second stage is bounded above (N +1)Mmax. In the EDF scheme (see Figure 3), every packet is assigned adeadline that is the departure time from the corresponding FCFSoutput-buffered switch. Packets are scheduled according to theirdeadlines in the central buffers. For the EDF scheme, there is noneed to implement the jitter control mechanism in the FCFS scheme.As such, average packet delay can be greatly reduced. However, asthere is no jitter control, one might need a larger resequencingbuffer than that in the FCFS scheme with jitter control. Since thefirst stage is the same as that in the FCFS scheme, both the delayand the buffer size of the load-balancing buffer are still boundedby N Lmax. Moreover, we show the followingtheorem for the EDF scheme. Its proof is given in the full report[3]. Theorem 2 Suppose that all the buffers are empty attime 0. Then the followings hold for the EDF scheme. (i) The end-to-end delay for a packet through our switch withmulti-stage buffering is bounded above by the sum of the delaythrough the corresponding FCFS output-buffered switch and N(Lmax + Mmax). (ii) The resequencing-and-output buffer at an output port ofthe second stage is bounded above N(Lmax+Mmax). Computing the departure times from the corresponding FCFSoutput-buffered switch needs global information of all the inputs.A simple way is to use the packet arrival times as deadlines. Thenthe EDF scheme based on arrival times yields the same departureorder except those packets that arrives at same time. Since thereare at most Mmax packets that can arrive at thesame time to an output port of the corresponding output-bufferedswitch, the end-to-end delay for a packet through the multi-stageswitch using arrival times as deadlines is bounded above by the sumof the delay through the corresponding FCFS output-buffered switchand N Lmax+(N+1)Mmax.Also, the resequencing-and-output buffer at an output port of thesecond stage in this case is bounded above N Lmax+ (N + 1)Mmax.

On the throughput of multicasting with incremental forward error correction

In this paper, we consider a multicasting model that uses incremental forward error correction (FEC). In this model, there is one sender and r/sup n/ receivers. The sender uses an ideal (n,n(1-p),np) FEC code to code a group of n(1-p) data packets with additional np redundant packets so that any set of n(1-p) packets received by a receiver can be used to recover the original n(1-p) data packets. Packets to the receivers are lost independently with probability q. For this model, we prove several strong laws of large numbers for the asymptotic throughput as n /spl rarr/ /spl infin/. The asymptotic throughput is characterized by the unique solution of an equation in terms of p, q, and r. These strong laws not only provide theoretical justification for several important observations made in the literature, but also provide insights that might have impact on future design of multicasting protocols.

Twister Networks and Their Applications to Load-Balanced Switches

Inspired by the recent development of optical queueing theory, in this paper we study a class of multistage interconnection networks (MINs), called {\em twister networks}. Unlike the usual recursive constructions of MINs (either by two-stage expansion or by three-stage expansion), twister networks are constructed {\em directly} by a concatenation of bipartite networks. Moreover, the biadjacency matrices of these bipartite networks are sums of subsets of the powers of the circular shift matrix. Though MINs have been studied extensively in the literature, we show there are several {\em distinct} properties for twister networks, including routability and conditionally nonblocking properties. In particular, we show that a twister network satisfying (A1) in the paper is routable, and packets can be self-routed through the twister network by using the

Generalized dynamic frame sizing algorithm for finite-internal-buffered networks

\cal C

Using Banyan Networks for Load-Balanced Switches with Incremental Update

-transform developed in optical queueing theory. Moreover, we define an

Load-balanced Birkhoff-von Neumann switches and fat-tree networks

N

Maximizing throughput in wireless networks with finite internal buffers

-modulo distance and use it to show that a twister network satisfying (A2) in the paper is conditionally nonblocking if the

A Universal Stabilization Algorithm for Multicast Flows with Network Coding

N