Approximate Computation of Loss Probability for Queueing System with Capacity Sharing Discipline

TL;DR

This paper develops an approximate formula for calculating the loss probability in a multi-channel queueing system with capacity sharing, considering different request types and dynamic service rates, validated against exact Markov chain solutions.

Contribution

It introduces a novel approximation method for loss probability in capacity sharing queueing systems with multiple request types and variable service rates.

Findings

Approximate loss probability formula closely matches exact values.

The method effectively handles systems with capacity sharing and diverse request types.

Validation shows high accuracy of the approximation compared to Markov chain solutions.

Abstract

A multi-channel queueing system is considered. The arriving requests differ in their type. Requests of each type arrive according to a Poisson process. The number of channels required for service with the rate equal to 1 depends of the request type. If a request is serviced with the rate equal to 1, then, by definition, the length of the request equals to the total service time. If at arrival moment, the idle channels is sufficient, then the arriving request is serviced with the rate 1. If, at the arrival moment, there are no idle channel, then the arriving request is lost. If, at arrival moment, there are idle channels but the number of idle channels is not sufficient for servicing with rate 1, then the request begins to be in service with rate equal to the ratio of the number of idle channels to the number of the channels required for service with the rate 1. If a request is serviced with a rate less than 1 and another request leaves the system, then the service rate increases for the request in consideration. Approximate formula for loss probability has been proposed. The accuracy of approximation is estimated. Approximate values are compared with exact values found from the system of equations for the related Markov chain stationary state probabilities.