Approximations for moments of response times in time sharing queues with limited number of service positions

A time sharing system consisting of a single processor with a limited number of service positions are unlimited waiting line is studied, in equilibrium, under a FIFO waiting line regime. A spectral method is applied to the calculation of the conditional response time moments in the case of state dependent arrival and service rates. The numerical results show that for the M/M/1 case the expected conditional response time is approximately a straight line, when presented as a function of the processing requirement length, in the region of interest. Thus in that region the behavior of the subsystem consisting of the service positions is similar to that of a symmetric queue. Based on this observation a linear approximation for the expected conditional response time in the M/G/1 case is obtained. The linear approximation is then compared to simulated results.