Technical Note—Conditional Delays Measured in Events for the M/M/c Queue

In many queuing systems the state of the system is known at each arrival epoch. Given an arrival and given the state of the system, the question of what the probability is that the new arrival will be delayed less than t units of time is often posed. For queuing systems with Poisson arrivals, negative exponential service times and various queue disciplines, this may involve considerable computation. In this paper we develop, for the M/M/c queue, recursive relations for calculating the conditional delays where t is measured in events rather than in units of time. These calculations are often simple to perform, even for some queuing models where the delay in units of time has not yet been obtained in closed form.