Transient Behavior of the M/M/1 and RBM

This memorandum is the fourth in a series investigating the time-dependent behavior of queues and related stochastic models. The principal goals are: (1) to obtain easily comprehensible closed-form approximate descriptions of the distribution and the moments of the number of customers in the system as functions of time and (2) to develop structural theorems that provide improved understanding. This memorandum focuses on the M/M1 queue starting with customers in the system. (The arrival and service rate in the M/M/1 model do not vary over time). Many of the results can be regarded as extensions of the Laplace transform description due to N.T.J. Balley (1954, 1957).