Generalized product form solutions for multiserver centres with concurrent classes of customers

Abstract

The MultiServer centre with Concurrent Classes of Customers (MSCCC) is a service centre consisting of B parallel identical exponential servers. The customers requesting service at the MSCCC centre belong to K classes. Customers arriving at the MSCCC centre are queued in the order of their arrival. A customer from class k will go into service at the MSCCC centre provided that one of the B servers is free and that at most nk - 1 other class k customers are in service at the MSCCC centre. Thus maximally nk class k customers may be concurrently in service at the MSCCC centre. The MSCCC centre can be applied to model resource contention in multiprocessing systems where K resources are accessed via B data paths: a process can access resource k if a data path is available and at most nk - 1 other processes are in service and are using resource k. This paper defines the MSCCC centre and presents several examples of computer (sub)systems that can be modelled using the MSCCC centre. The MSCCC centre is shown to satisfy local balance: therefore a queueing network consisting of BCMP and MSCCC centres has a product form solution. The Joint Probability Distribution (JPD) for a queueing network consisting of several BCMP centres and one MSCCC centre is derived. Aggregation techniques are used to reduce the JPD to a computationally tractable form. A Mean Value Analysis (MVA) algorithm is presented for calculating the closed chain performance measures at the MSCCC centre. Finally, MVA methods are used to calculate the open chain performance measures at the MSCCC centre.