A matrix analytic solution to a hysteretic queueing system with random server capacity
A matrix analytical approach is used to obtain steady-state distribution of the queue size in the hysteretic model with random server capacity. This is an M/G/1-type system combining \(r\)-quorum discipline and \(N\)-policy: the server is turned on when queue length reaches a level \(N\) and off when he terminates with \(r\) or less customers in the system. The underlying embedded Markov chain is obtained as the queue size process at the departure epochs (of a batch of customers). At that the new batch size is \(r\) if the queue size is not less than \(r\), otherwise the server is turned on when queue size reaches a level \(N\) (not less than \(r\)) and then batch size becomes random (and its service time depends on the number of customers in the batch). Instead of the traditional tedious approach based on moment generating functions and Laplace-Stieltjes transform, the author uses the matrix analytical approach to establish a stability condition and to compute the steady-state distribution. The approach is illustrated by a numerical example.
scientific article; zbMATH DE number 1860850
scientific article