Closed Finite Queuing Networks with Time Lags

Consider a closed finite queuing system in which m units move among stations 1, 2, …, N ( N ≧ 1). A unit that completes service at station i moves to station j ( i , j = 1, 2, …, N ) with probability e j ∣ i (∑ j = N j =1 e j ∣ i =1, i = 1, 2, …, N ) where it either commences service immediately if there is a server available, or queues to await service. The time taken for a unit to move from station i to station j is a random variable with distribution function G i,j (⋯). At each station, the service times are independent and exponentially distributed, with the instantaneous service rate at a station being an arbitrary function of the number of units at that station. The positions of the units enroute between stations, measured in time units, are introduced as supplementary variables yielding a Markov process with states having a combination of discrete and continuous components. The integro-differential difference equations for the state variables are derived, and the steady-state solution is determined. The marginal joint distribution for the number of units at each station is found, and shown to depend, not on the distributions of the time lags, but rather on some linear combination of the means of the time lag distributions. An approximate technique is outlined for determining the absolute probability distribution of the number of units at a station, and some of the limitations of this method are discussed.