Heuristic performance model of optical buffers for variable length packets

Optical switching (optical packet switching, optical burst switching, and others) provides alternatives to the current switching in backbone networks. To switch optically, also packet buffering is to be done optically, by means of fiber delay lines (FDLs). Characteristic of the resulting optical buffer is the quantization of possible delays: Only delays equal to the length of one of the FDLs can be realized. An important design challenge is the optimization of the delay line lengths for minimal packet loss. To this end, we propose a heuristic based on two existing queueing models: one with quantization and one with impatience. Combined, these models yield an accurate performance modeling heuristic. A key advantage of this heuristic is that it translates the optical buffer problem into two well-known queueing problems, with accurate performance expressions available in the literature. This paper presents the heuristic in detail, together with several figures, comparing the heuristic’s output to existing approaches, validating its high accuracy.

The first author is a Postdoctoral Fellow with the Research Foundation Flanders (FWO-Vlaanderen), Belgium. This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

Authors and Affiliations

1. Department of Telecommunications and Information Processing, Ghent University (UGent), St.-Pietersnieuwstraat 41, 9000 , Ghent, Belgium

   W. Rogiest, K. Laevens, S. Wittevrongel & H. Bruneel

Corresponding author

Correspondence to W. Rogiest.

Appendix: the discrete-time case

While the results of the body of this contribution apply only to a continuous-time setting, this Appendix shows that minor changes suffice to make it applicable to a discrete-time setting. As in the analysis for continuous time, we assume a degenerate buffer setting, with line lengths equal to multiples of the granularity \(D\). We specifically focus on the case of an M/M/1 optical buffer, now in discrete time.

In a discrete-time setting, events take place synchronously, at the beginning of time slots. Therefore, all time-related variables and performance measures are expressed as multiples of the slot length, and for example, inter-arrival times and packet sizes take on only strictly positive integer values, contained in \(\mathbb N _{0}\). The slot length may be arbitrary and is therefore not mentioned explicitly in this Appendix.

A first point of attention is that in discrete time, the relation between queues with quantization and queues with impatience is tighter than in the continuous-time case. While in continuous time, a queue with impatience was obtainable from the optical buffer by letting \(D\rightarrow 0\), it suffices to set \(D=1\) in an optical buffer model to obtain a discrete-time model for queues with impatience. This has large impact on the way in which impatience is modeled. In the present context, it suffices to set \(D=1\) in all discrete-time results of [14], to obtain exact results for a general GI/G/1 system with deterministic impatience in discrete time. Contrasting this with results for continuous time, it is remarkable that even the less general M/G/1 system in continuous time of [19] requires numerical approximations in order to obtain results. Apparently, assuming a discrete-time setting somewhat simplifies the queueing problem with impatience.

Further, discrete-time is also the time setting studied in [17], where a general impatience distribution with upper bound \(r\) is assumed. This allows to handle a deterministic impatience distribution, by setting \(r=\tau +1\). (The term +1 is to be introduced to be compatible with the definition of \(r\) in [17]).

As for the traffic assumptions, we adopt the same indexing convention as for continuous time. For the arrival process, we assume a Bernoulli arrival process, which is the discrete-time counterpart of the Poisson arrival process. The inter-arrival times, a sequence of iid random variables, have a common geometric distribution with cumulative distribution function

with \(p\in [0,1]\). The latter probability is also the parameter of the geometric distribution and gives the probability of having an arrival in an arbitrary slot, relating to the mean value, as \(\text{ E }[T_{k}]=1/p\). The packet sizes again form a sequence of iid random variables with geometric distribution, now with common probability mass function \(b(n)=\text{ Pr }[B_{k}=n]\) and cumulative distribution function \(B(n)=\text{ Pr }[B_{k}\le n]=1-(1-f)^n, n\in \mathbb N \), with \(f\in [0,1]\), with mean value \(\text{ E }[B_{k}]=1/f\). As in continuous time, knowledge of the granularity \(D, T(n)\) and \(B(n)\) suffices as input for the analysis in discrete time.

For the modeling of impatience in discrete time, we rely on Sect. 3 of [17], reporting the probability that the age of the customer in service is zero (there denoted \(\hat{\pi }_{0}\)) equals

with \(\bar{p}=1-p\), and \(\bar{f}=1-f\). With now \(\text{ LP }=1-(1-\hat{\pi }_{0})/\rho \) and \(r=N+1\), one easily obtains

with \(\rho =p/f\). Note that this expression for the LP is tightly related to the one of continuous time (7). To see this, we rewrite the continuous-time expression for \(\rho \ne 1\) as

Now, a substitution similar to the one in continuous time is needed to yield correspondence,

completed with \(\tau =N+1\), to indeed obtain (9). The expression for \(\rho =p/f=1\) follows by taking the limit for \(p\rightarrow f\). The link between discrete time and continuous time is less intuitive at one point, since \(\tau \) is “virtually expanded” to \(N+1\) in discrete time, rather than \(N\). The latter, however, forms no stumbling block: It can be understood as (an indirect) result of the difference in the minimum of the inter-arrival times in discrete time and continuous time, 1 and 0, respectively.

Together with the expression for \(\rho _\mathrm{eq}\) in discrete time, whose derivation is straightforward, (9) can be used for an approximate modeling of optical buffers. This is not treated further here, since the accuracy and the obtained associated figures are very similar to the continuous-time case.

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