IntroductiontoManagementScience-California.ppt

A Basic Queueing System Herr Cutter’s Barber Shop Herr Cutter is a German barber who runs a one-man barber shop. Herr Cutter opens his shop at 8:00 A.M. The table shows his queueing system in action over a typical morning. Arrivals The time between consecutive arrivals to a queueing system are called the interarrival times. The expected number of arrivals per unit time is referred to as the mean arrival rate. The symbol used for the mean arrival rate is l Mean arrival rate for customers coming to the queueing system where l is the Greek letter lambda. The mean of the probability distribution of interarrival times is 1 / l Mean interarrival time Most queueing models assume that the form of the probability distribution of interarrival times is an exponential distribution. Evolution of the Number of Customers The Exponential Distribution for Interarrival Times Properties of the Exponential Distribution There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice. For most queueing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly. Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same. The only probability distribution with this property of random arrivals is the exponential distribution. The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lack-of-memory property. The Queue The number of customers in the queue or queue size is the number of customers waiting for service to begin. The number of customers in the system is the number in the queue plus the number currently being served. The queue capacity is the maximum number of customers that can be held in the queue. An infinite queue is one in which, fo