RECAL is a new recurrence relation for calculating the partition function and various queue length moments for closed, product form networks. In this paper we discuss a number of the issues involved in the software implementation of RECAL on both sequential computers and parallel, shared memory computers. After a brief description of RECAL, we describe software implementing RECAL on a sequential computer. In particular, we discuss the problems involved in indexing and data storage. Next we describe code implementing RECAL on a parallel, shared memory computer. Special attention is given to designing a special buffer for temporary data storage and several other important features of the parallel code. Finally, we touch on software for serial and parallel implementations of a tree algorithm for RECAL.

Striking progress has been made recently in obtaining expressions for the sojourn time distribution function (STDF) of a job at a c -server, first-come, first-serve (FCFS) center in a closed, product-form queuing network. These results have more recently been extended, and expressions have been obtained for the joint distribution function (DF) of the sojourn times of a job at a sequence of single-server, FCFS centers lying on an “overtake-free” path. However, these formulas present considerable computational problems in the case of large, closed queuing networks. In this paper, asymptotic techniques developed by Mitra and McKenna for the calculation of the partition function of large, product-form closed queuing networks are applied to the sojourn time problem. Asymptotic expansions are obtained for the STDF of a job at c -server, FCFS center in closed, product-form queuing networks. Similar expansions are obtained for the joint DF of the sojourn times of a job at a sequence of single server, FCFS centers lying on an “overtake-free” path. In addition, integral expressions are obtained for the STDF of a job at a single server, FCPS center in a closed, product-form queuing network in which all the centers are load independent. These integral expressions also yield useful asymptotic expansions. Finally, integral expressions are also obtained for the joint DF of the sojourn times of a job at the centers of an “overtake-free” path in such a network.

A method is presented for calculating the partition function, and from it, performance measures, for closed Markovian stochastic networks with queuing centers in which the service or processing rate depends on the center's state or load. The analysis on which this method is based is new and a major extension of our earlier work on load-independent queuing networks. The method gives asymptotic expansions for the partition function in powers of 1/ N , where N is a parameter that reflects the size of the network. The expansions are particularly useful for large networks with many classes, each class having many customers. The end result is a decomposition by which expansion coefficients are obtained exactly by linear combinations of partition function values of small network constructs called pseudonetworks. Effectively computable bounds are given for errors arising from the use of a finite number of expansion terms. This method is important because load dependence is at once an essential element of sophisticated network models of computers, computer communications, and switching, teletraffic, and manufacturing systems, and the cause of very intensive computations in conventional techniques. With this method, very large load-dependent networks can be analyzed, whereas previously only small networks were computationally tractable.

In this paper a method for calculating the first and higher order moments of the queue lengths for each customer class at each node in a dosed Markovian queuing network is presented.The method is based on asymptotic expansions in powers of N -~ for the moments of interest.N is a parameter that reflects the size of the network.The derivation of the asymptotic expansions presented here is based on the techniques developed by us earlier to get asymptotic expansions in powers of N -a of the mean utilization of each processor by each class of customers.These expansions are valid in the "normal usage" case in which none of the processors are too heavily utilized.These expansions are particularly useful m the ease of large networks with many classes of customers, each class having many customers.With these expansions, we are able to analyze very large networks that earlier were computationally intractable.This is important, because dosed Markovian queuing networks have emerged as one of the most important tools for modeling computer systems, computer communication systems, on-line computer networks and other real-time computer-based system.Previous to the work reported here, their use had been restricted to relatively small networks since their use in large networks involved intractably large calculations.