Joint Stationary Distribution Of Queues In Homogenous M|M|3 Queue With Resequencing

Abstract

Resequencing issue is a crucial issue in simultaneous processing systems where the order of customers (jobs, units) upon arrival has to be preserved upon departure. In this paper stationary characteristics of M/M/3/∞ queueing system with reordering buffer of infinite capacity are being analyzed. Noticing that customer in reordering buffer may form two separate queues, focus is given to the study of their size distribution. Expressions for joint stationary distribution are obtained both in explicit form and in terms of generating functions. Numerical example is presented.

Estimation of Network Disordering Effects by In-depth Analysis of the Resequencing Buffer Contents in Steady-state

“…Customers, which violated the arrival order are kept in the resequencing buffer (RB) of infinite capacity before each of them can leave the system. As it was noticed in [16], in such M|M|N|∞ resequencing queue with N > 2 servers, the resequencing buffer can be thought of either as a single queue, where all customers which violated arrival order reside together (Fig. 1a) or as a collection of several separate interconnected queues (Fig.…”

“…The analysis of the joint stationary distribution of number of customers even in simple cases with Poisson flow and homogeneous exponential servers turns out to be a challenging task. In [16] for M/M/3/∞ queue followed with infinite resequencing buffer one obtains expressions for joint stationary distribution of number of customers in buffer and servers, and number of customers in each of two queues in resequencing buffer both in explicit form and in terms of generating functions. In [17] for M/M/N/∞ queue followed with infinite resequencing buffer there was obtained algorithm for recursive computation joint stationary distribution of number of customers in buffer and servers, and sum of number of customers in two, three, .…”

“…Here it is shown that in the general case N > 3 the joint stationary distribution can be computed recursively. The special case of this methodology has already been used in [16]. The authors note that the joint distribution for the general case can be also obtained algorithmically in terms of the generating functions (as it is shown in [18]), but that results are, as usual, hardly applicable for the computation of the joint distribution itself.…”