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We compute the stationary performance metrics of a single server $M^X/G/1$queue under a class of generalized processor-sharing scheduling policies thatare proposed by Grishechkin. This class of processor-sharing policies allowservice capacities to be allocated to jobs based on the amount of service theyattained. In [11], Grishechkin derives an integral equation that is satisfiedby the Laplace transform of the stationary performance metrics under thesepolicies. Our main focus in this paper is to derive the solution to thisintegral equation. This is achieved through a series of transforms that convertthe integral equation into a more tractable form, then solve it. Then we deriveapproximations to the density functions of the performance metrics throughinverting the Laplace transforms. In the last part of the paper, we apply ourresults to some well-known scheduling policies that are either a special caseof the Grishechkin processor-sharing policies or can be treated as the limitsof a sequence Grishechkin processor-sharing policies. These examples includethe egalitarian processor-sharing policy, the discriminatory processor-sharingpolicy, shortest residual processing time first rule, foreground-backgroundpolicy and the time function scheduling policy.

Stationary Performance Analysis of Grishechkin Processor-Sharing Queues: An Integral Equation Approach