Jay Cheng

A Necessary and Sufficient Condition for the Construction of 2-to-1 Optical FIFO Multiplexers by a Single Crossbar Switch and Fiber Delay Lines

In this paper, we prove a necessary and sufficient condition for the construction of 2-to-1 optical buffered first-in-first-out (FIFO) multiplexers by a single crossbar switch and fiber delay lines. We consider a feedback system consisting of an (M+2)times(M+2) crossbar switch and M fiber delay lines with delays d1,d2,...,dM. These M fiber delay lines are connected from M outputs of the crossbar switch back to M inputs of the switch, leaving two inputs (respectively, two outputs) of the switch for the two inputs (respectively, two outputs) of the 2-to-1 multiplexer. The main contribution of this paper is the formal proof that d1=1 and di les di+1 les 2d i, i=1,2,...,M-1, is a necessary and sufficient condition on the delays d1,d2,...,dM for such a feedback system to be operated as a 2-to-1 FIFO multiplexer with buffer Sigmai=1 Mdi under a simple packet routing policy. Specifically, the routing of a packet is according to a specific decomposition of the packet delay, called the C- transform in this paper. Our result shows that under such a feedback architecture a 2-to-1 FIFO multiplexer can be constructed with M=O(log B), where B is the buffer size. Therefore, our construction improves on a more complicated construction recently proposed by Sarwate and Anantharam that requires M=O(radicB) under the same feedback architecture (we note that their design is more general and works for priority queues)

Capacity of Nakagami-q (Hoyt) fading channels with channel side information

In this paper, we derive closed-form expressions for the channel capacity of a diversity reception system with maximal-ratio combining (MRC) or postdetection equal-gain combining (EGC) over a slowly varying Nakagami-q (Hoyt) fading channel under different adaptive transmission policies when the channel side information is available at the transmitter and the receiver. We consider the situation that the branches are independent and identically distributed (i.i.d.). We first derive three different canonical-form expressions for the probability density function (PDF) of the instantaneous combiner output signal-to-noise ratio (SNR) as a weighted sum of gamma PDFs. Then the channel capacity is obtained in different canonical forms as a weighted sum of channel capacities for single-branch systems in gamma fading environments.

A simple proof for the constructions of optical priority queues

Abstract Constructions of optical queues by optical Switches and fiber Delay Lines (SDL) have received a lot of attention lately. In this short paper, we provide a simple proof for the construction of a priority queue with a switch and fiber delay lines in Sarwate and Anantharam (Queueing Syst. Theory Appl. 53, 115–125, 2006). Our proof not only gives the insights needed to understand why such a construction works, but also leads to a more general result that recovers the result in Sarwate and Anantharam (Queueing Syst. Theory Appl. 53, 115–125, 2006) as a special case.

Using a Single Switch with O(M) Inputs/Outputs for the Construction of an Optical Priority Queue with O(M 3 ) Buffer

In this paper, we consider the construction of an optical priority queue with a single (M+1)times(M+1) switch and M fiber delay lines. The M fiber delay lines are connected from M outputs of the switch back to M inputs of the switch, leaving one input (resp. output) of the switch for the input (resp. output) of the priority queue. It was known that with an appropriate choice of the lengths of the delay lines, such a construction can be used for exact emulation of an optical priority queue with O(M2) buffer size. In this paper, we show that the buffer size can be further extended to O(M3) using the same construction. The improvement relies on establishing a partial ordering for all the packets stored in the delay lines.

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

One of the most popular approaches for the constructions of optical buffers needed for optical packet switching is to use switched delay lines (SDL). Recent advances in the literature have shown that there exist systematic SDL construction theories for various types of optical buffers, including first-in first-out (FIFO) multiplexers, FIFO queues, priority queues, linear compressors, nonovertaking delay lines, and flexible delay lines. As parallel FIFO queues with a shared buffer are widely used in many switch architectures, e.g., input-buffered switches and load-balanced Birkhoff-von Neumann switches, in this paper we propose a new SDL construction for such queues. The key idea of our construction for parallel FIFO queues with a shared buffer is two-level caching, where we construct a dual-port random request queue in the upper level (as a high switching speed storage device) and a system of scaled parallel FIFO queues with a shared buffer in the lower level (as a low switching speed storage device). By determining appropriate dumping thresholds and retrieving thresholds, we prove that the two-level cache can be operated as a system of parallel FIFO queues with a shared buffer. Moreover, such a two-level construction can be recursively expanded to an n-level construction, where we show that the number of 2times2 switches needed to construct a system of N parallel FIFO queues with a shared buffer B is O((NlogN)log(B/(NlogN))), for NGt1. For the case with N=1, i.e., a single FIFO queue with buffer B, the number of 2times2 switches needed is O(logB). This is of the same order as that previously obtained by Chang We also show that our two-level recursive construction can be extended to construct a system of N parallel last-in first-out (LIFO) queues with a shared buffer by using the same number of 2times2 switches, i.e., O((NlogN)log(B/(NlogN))), for NGt1 and O(logB) for N=1. Finally, we show that a great advantage of our construction is its fault tolerant capability. The reliability of our construction can be increased by simply adding extra optical memory cells (the basic elements in our construction) in each level so that our construction still works even when some of the optical memory cells do not function properly

Feedforward SDL Constructions of Output-Buffered Multiplexers and Switches with Variable Length Bursts

In this paper, we study the problem of exact emulation of two types of optical queues: (i) N-to-1 output-buffered multiplexers with variable length bursts, and (ii) N times N output-buffered switches with variable length bursts. For both queues, the delay of a packet (in a burst) is known upon its arrival. As such, one can emulate such queues by finding a delay path that yields the exact delay for each packet. For emulating the delay of a packet in such queues, in this paper we consider a multistage feedforward network with optical crossbar switches and fiber delay lines (SDL). For any fixed delay d, there exist multiple delay paths in such a network. A delay path is feasible if it satisfies the following three constraints: (i) conflict constraint: no more than one packet can be scheduled at the same input/output ports of each crossbar switch at the same time, (ii) causality constraint: no packet can be scheduled before its arrival, and (iii) strong contiguity constraint: packets in the same burst should be routed through any fiber delay lines contiguously. By the worst case analysis, we find sufficient conditions for the numbers of delay lines needed in each stage of such a feedforward network to achieve exact emulation of both queues. For N-to-1 output-buffered multiplexers, our sufficient conditions are also necessary when each burst contains exactly one packet. By computer simulation, we also show that the number of delay lines in each stage can be greatly reduced due to statistical multiplexing gain.

Multistage Constructions of Linear Compressors, Non-Overtaking Delay Lines, and Flexible Delay Lines

Queueing theory is generally known as the theory to study the performance of queues. In this paper, we are interested in another aspect of queueing theory, the theory to construct queues via switched delay lines. We consider three types of discrete-time queues: linear compressors, non-overtaking delay lines and flexible delay lines. These three types of queues correspond to certain conditional nonblocking switches and (strict sense) nonblocking switches in switching theory. Analogous to their counterparts in switching theory, there exist multistage constructions for these three types of queues. Specifically, we develop a two-stage construction of a linear compressor and a three-stage construction of a non-overtaking delay line. Similarly, there is a three-stage construction of a flexible delay line. Moreover, a flexible delay line can also be constructed by a layered Cantor network.

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.

Performance analysis for MRC and postdetection EGC over generalized gamma fading channels

In this paper, we provide a unified analysis of average symbol error probability (SEP) for a diversity system over generalized gamma fading channels, which is a generalization of Rayleigh, Nakagami, and Ricean fading channels. We consider independent generalized gamma fading on the diversity branches and derive different closed-form expressions of the average SEP for a general class of M-ary modulation schemes (including MPSK, MQAM, BFSK, and MSK) with maximal-ratio combining (MRC) and for M-ary orthogonal FSK with postdetection equal-gain combining (EGC). The results apply to the situations where some branches are Nakagami faded and the others are Ricean faded. Furthermore, the results are applied to obtain closed-form expressions of the average SEP for the cases of arbitrarily correlated and not necessarily identically distributed Nakagami and Ricean faded branches with the help of virtual branch technique by Win et al. Our approach provides a canonical structure for the average SEP as a weighted sum of elementary closed-form expressions, which are the closed-form expressions for the average SEP of a diversity system in independent and identically distributed (i.i.d.) Nakagami fading environments.

Computable exponential bounds for intree networks with routing

In this paper, we refine the calculus proposed previously by Chang et al. (1994). The new calculus, including network operations for multiplexing, input-output relation, and routing, allows us to compute tighter exponential bounds for the tail distributions of queue lengths in intree networks with routing. In particular, if external arrival processes and routing processes are either Markov arrival processes or autoregressive processes, the stationary queue length at a local node is stochastically bounded above by the sum of a constant and an Erlang random variable. The decay rate of the Erlang random variable is not greater than (in some cases equal to) the decay rate of the tail distribution of the stationary queue length. The number of stages of the Erlang random variable is the number of external arrival processes and routing processes contributing to its queue length. For the single queue case, both the lower and upper-bounds are derived.