Asymptotic Analysis of the Time Dependent M/M/1 Queue

Using operator analytic techniques, we develop a nonstationary Markovian queueing theory starting with the M(t)/M(t)/1 queue. We employ an asymptotic approach quite different from the usual large time analysis. Instead, we uniformly accelerate the queue length process. That is, we divide the arrival and service rate by a common parameter ϵ. Then, for a fixed time interval, we consider the asymptotics for the distribution, mean, and variance of the queue length process as ϵ goes to zero. The effects of ϵ can be quite different for the given time interval. This gives us a dynamic asymptotic behavior for the queue length process. We can formulate a time dependent traffic intensity parameter that determines when the process is asymptotically stable and when it is asymptotically unstable.