Queueing Systems with Daily Cycles and Stochastic Demand with Uncertain Parameters

Andrew M. Ross

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Telecommunications systems see large daily cycles in demand, and when they are designed the demand they will face is uncertain. We look at three typical telephone-related queueing systems from the Internet-access and call-center industries.

Our first model considers the cost of providing service via sub-contractors. We present a cost model for taking Internet dial-up traffic that varies by time-of-day and splitting it between two large modem banks, one of which charges by the hour, the other charges for the peak. To study if the possible savings are enough to make the effort worthwhile, we formulate a ``perfect information'' Integer Program that is equivalent to a network flow problem. In the stochastic case, we use a Normal approximation, and develop a square-root-type rule to set the ceiling in the homogeneous case. We also use simulation to determine an optimal ceiling when we cannot route individual calls precisely.

In our second model, we estimate the number of modems and characteristics of the opposing traffic stream in a partially observed Erlang loss system (individual modem bank). Using detailed sample-path data, we construct a Maximum Likelihood Estimator that makes good use of the data, but is slow to evaluate. As an alternative, we present an estimation system based on traffic data summarized by hour. This estimation system is much faster, and tends to produce good lower bounds on the size of the system and competing traffic.

In our third model, we consider designing a call center system when we cannot predict the amount of traffic accurately. We try to decide on a number of servers and number of extra buffer spaces (number of calls that can wait on-hold), given a prior distribution on the input traffic. The systems are designed to satisfy upper bounds on the probabilities of blocking or delay. For infinite-buffer systems, we find that a previously published approximation tends to under-estimate the needed number of servers. For finite-buffer systems with known traffic, we adapt the approximation to handle the trade-off between adding servers and adding buffers.