242 IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 22, NO. 4, FEBRUARY 15, 2010 Exact Optimization Method for an FDL Buffer With Variable Packet Length Wouter Rogiest and H. Bruneel Abstract—This letter proposes an exact and simple method to optimize the fiber lengths of fiber delay line buffers in an asyn- chronous network with variable packet length. Existing algorithms required considerable calculation, which can be avoided by using the closed-form loss probability expressions we obtained, valid for general traffic conditions. Index Terms—Buffers, communication switching, computer net- work performance, optical fibers. I. INTRODUCTION K EEPING up with increasing bandwidth demand urges for a faster backbone network, possible by making use of the new solutions that optical technology provides. This vision re- flects in optical packet switching [1] and optical burst switching [2], which allow us to forward data in an optical form. The in- volved optical buffers allow us to delay packets (or bursts) to ensure that they do not overlap when they are transmitted over the output link. A common implementation of an optical buffer is by means of fiber delay lines (FDLs), which can be used in a feed-forward setup, each line providing the delay that corre- sponds to its length. This implementation is counterpart to the feedback setup [3]. The lengths in a feed-forward setup are usu- ally assumed multiples of a basic delay unit called granularity [4]. In a network with fixed-length packets, a natural choice is to let match that packet length, although this is not necessarily the optimal choice [5], [6]. Of particular interest to this work is the case of asynchronous variable length packets, where such a “natural choice” is not readily available, and the line length optimization problem involves both the buffer size and traffic load. The often-cited letter [4] accounts for the prime contribution treating optical buffering, with recent applications relying on it, see for instance [7]. However, the solution of [4] (and its extended version, [8]) only provided an approximate solution, that is inaccurate in some cases, as discussed below. An enhanced approach with Markov chains enabled accurate numerical results [9], but still requires considerable numerical calculation to obtain those results. Opposed to this, we obtained exact closed-form expressions for the loss and waiting time probabilities, valid for any parameter setting at once, without requiring numerical Manuscript received July 08, 2009; revised October 09, 2009; accepted De- cember 02, 2009. First published January 12, 2010; current version published January 27, 2010. W. Rogiest is with the SMACS Research Group, TELIN Department, Ghent University (UGent), B-9000 Ghent, Belgium, and also with Bell Laboratories, Alcatel-Lucent Bell NV, B-2018 Antwerpen, Belgium (e-mail: wouter.rogiest@UGent.be) H. Bruneel is with the SMACS Research Group, TELIN Department, Ghent University (UGent), B-9000 Ghent, Belgium. Digital Object Identifier 10.1109/LPT.2009.2038237 means. In fact, the expressions presented here greatly simplify the optimization of optical switching mechanisms, such as the dynamic burst length adjustment mechanism proposed in [7]. The latter mechanism still required a separate and involving algorithm for the calculation of the optimal granularity, whereas the presented expressions, being exact and closed-form, allow us to obtain it right away. To the best of our knowledge, for FDL buffers of finite size, reference [6] (fixed length packets) and the current contribution (variable length packets) are the first to report such expressions. II. FDL BUFFER SETTING Since the setting studied in [4] is exactly the focus of the cur- rent contribution, we adopt all notation coined there ( , , , , , , ). The optical buffer is dedicated to a single output wavelength (or, single server), and is located at the output of the switch. In a buffer with FDLs, delay line with index (0 to ) provides a delay of s, resulting in a max- imum delay of s. Buffer control exercises a first-come-first-served scheduling discipline: in case an arriving packet finds an idle system, the packet is served immediately; in case of a nonidle system, the arriving packet is delayed for at least the time needed for the one-but-last packet to leave the buffer. Given the limited set of possible delays, the packets are not delayed for the requested time but somewhat more, so as to match the smallest multiple of . Intuitively, it can be un- derstood that minimizing allows us to minimize this capacity waste, during which the outgoing channel remains unused, de- spite of work still being present in the buffer. However, since the maximum delay is also a function of ( ), an optical buffer, like any buffer, provides better loss perfor- mance as its size increases, which urges us to maximize . An apparent trade-off occurs, showing that minimal packet loss is obtained for some intermediary value of that we call . III. MARKOV CHAIN OF WAITING TIMES While the buffer setting is identical to that of [4], the analysis follows different lines, by focusing on the Markov chain of the waiting times of arriving packets. Numbering packets in the order of their arrival by index , an arbitrary packet either experiences a waiting time in the buffer, , or is discarded if the buffer is full and the requested delay exceeds . Both the packet sizes and interarrival times constitute a series of i.i.d. random variables with negative exponential distribution. With packet , we associate a packet size with mean value . The interarrival time is defined as the time between the th packet and the th, with mean value . Now, the system’s evolution can be captured in terms of and , by considering the acceptance or loss of packet , conditioned on the acceptance or loss of packet . 1041-1135/$26.00 © 2010 IEEE