A Simple Approximation to the Average Queue Size in the Time-Dependent M/M/1 Queue

The time-dependent equations for the M/M/1 queue can be reduced to a single equation for the expected queue size, but the equation is dependent on P 0 ( t ), the probability of no jobs in the system. An exact equation for the behavior of P 0 ( t ) under special conditions is derived and an approximation relating P 0 ( t ) to Q ( t ), the expected queue size at time t , is derived for the case when the change in queue size is slow compared to the service rate. It is found that the approximation affords a significant improvement over the use of a steady state approximation to the time-dependent queue and is simpler to use than the exact equations.