Approximation Techniques for Queueing Systems with Finite Waiting ,room

The paper presents a short overview of two approximation techniques that may be appropriate to estimate performance measures of queueing systems withnon-Poisson offered traffic and finite queue. The techniques evolved from similar techniques originally designed for pure loss systems [1,3J, recently generalized to systems with infinite queue [4J. Consider the queueing system GI/M/N/N+Q, in which renewal traffic is offered to N exponential servers and a finite number Q of waiting places. Customers that arrive when the total system capacity N+Q is fully occupied are lost from the system and do not return. We report a study on approximation techniques to estimate the performance measures of the GI/M/N/N+Q-queueing system, viz., the blocking probability B, the waiting probability Wand the mean waiting time T, based on non-lost arrivals. In the approximation models offered traffic is supposed to be su'lHiciently described by its mean M and its peakedness Z (Z=V/M, where V represents the variance). The approximations are such that the performance measures of the original GI/M/N/N+Qsystem are expressed in terms of the corresponding measures B', W' and T' of an adjoint system with Poisson input, viz., the Markovian queue M/M/N'/N '+Q' (quantities in the adjoint system are indicated with ' in the sequel). The adjoint system provides us with exact formulas E'2 J that can be advantageously expressed in terms of Erlang loss functions. Fast algorithms are available to compute the Erlang formula for integral as well as nonintegral values of N'. In order to evaluate the approximations, exact analysis of GI/M/N/N+Q [5J , in general complicated, has been performed for hyperexponentially and gamma distributed interarrival timffiin case of peaked (Z>I) and smooth (~i) offered traffic, respectively. The perfomance measures 'B', W' and T' of the adjoint system, with offered Poisson traffic M' , are: ( 1) B' =b (M', N~ Q'), W=w (~, N', Q~, T~t (M~ N', cb, where the functi ons b, wand t are known [2J . First,we introduce an approximation technique that is a new application of the decomposition method as explained in [3J . The approximation formulas are ('decom' in the example): (2) B Z • B' Z • b (M' ,N' , Q ' ) , (3) W ~ a .Z. W' =a.Z.w(M' ,N' ,Q'), (4) T ~ Z.T' = Z.t(M' ,N'~Q'), where (5) M'=V, N'=N-M+V, Q'=Q, a =N/N'. Formulas (2) and {4) follow from the rather simple decomposition argument [3,4J. The