Delay analysis of a queue with general service demands and phase-type service capacities

We present the analysis of a non-classical discrete-time queueing model where customers demand variable amounts of work from a server that is able to perform this work at a varying rate. The service demands of the customers are integer numbers of work units. They are assumed to be independent and identically distributed (i.i.d.). The service capacities, i.e., the numbers of work units that the server can process in the consecutive slots, are also assumed to be i.i.d. and have a rational probability generating function (pgf). Finally, the numbers of customer arrivals in each slot are i.i.d. as well. We analyze this model analytically using contour integration. Our main result is an expression for the pgf of the customer delay in steady state, from which expressions for the moments of the delay can be derived