Application of Utilization Rate Approach for the Evaluation of Effective Numbers of Servers for A Parallel System of Single-Server Queues

Abstract

Prospective service delivery and manufacturing firms as the case may be, are desirous of knowing the number of servers and service delivery operating equipment to be installed in their proposed facilities, to ensure adequate service delivery. Perhaps, this underlining interest may not be unconnected with the fact that finite facility space and limited income constraints probably will not allow for infinite severs and manpower. In this paper, the M/M/K (an exponentially distributed interarrival, service time and multiple servers) parallel system of single-sever queues model is modified and deployed to analyse the services of a petroleum product loading depot. The adequacy or otherwise of the loading process is measured by a stability equation developed from the utilization rate equation. The constraints of the stability equation required to be satisfied to ensure effective service delivery (bounded queues and decongested facility) establish a range of new queue parameters. The least mean service rate value within this range can be evaluated with a server equation to produce the minimum number of effective servers, required by any firm that will yield a utilization factor of less than one. A numerical evaluation of the stability equation and the server equation is carried out with primary data sourced from the Warri refinery depot. It is found that the current operating system in the depot with nine parallel servers is ineffective with a utilization factor of 1.17, hence the resultant congestion in the depot. Upon satisfying the stability equations constraints, a new mean service rate value is estimated that gives a utilization factor of 0.97 less than one. The new service time value corresponds to a minimum effective number of eleven servers by the server equation, to give a utilization factor of 0.97. Better services with lower utilization rate values can be achieved by varying the mean service time along the established range of the stability equation’s constraint.