On the Growth of the Maximum Queue Length in a Stable Queue

Even for a stable one-server queue with relative traffic intensity less than unity, the maximum queue length observed will tend to infinity in probability as the length of time for which the system has been observed becomes large. It is important to have information on the behavior of the maximum queue length to enable estimation of the congestion likely to be encountered in a system operating over a long period. The present paper studies the stable GI/M/1 system, and, interpreting the results obtained rather loosely, one can say that the maximum queue length attained by the time the nth customer has arrived (n large) can be expected to be of the order of log n.