A clerk provides exponentially distributed service time to...

A clerk provides exponentially distributed service time to customers who arrive randomly at the average rate of 15 per hour. What average service time must the clerk provide in order that $90 \%$ of all customers will queue for service for a time not exceeding 12 minutes?

Key Concepts

M/M/1 Queue Model

The M/M/1 queue is a fundamental model in queueing theory that features a single server with both interarrival times and service times following an exponential distribution. This model provides a framework for analyzing system metrics such as waiting times and queue lengths under steady state conditions.

Waiting Time Distribution in Queues

In an M/M/1 queue, the distribution of waiting time for customers, excluding service time, is also exponential under equilibrium. By using the cumulative distribution function of this exponential waiting time, one can assess metrics such as the probability that a customer’s waiting time stays below a particular threshold.

Exponential Distribution

The exponential distribution is a continuous probability model commonly used to describe the time between independent, random events occurring at a constant average rate. Its key property—the memoryless property—implies that the chance of an event occurring in the next moment is independent of how much time has already passed.

Poisson Process for Arrivals

A Poisson process is a stochastic process that models a series of events occurring randomly over time at a constant average rate. This process is frequently used in queueing theory to represent the arrival of customers or jobs because arrivals are assumed to be independent and memoryless.

Traffic Intensity

Traffic intensity, defined as the ratio of the arrival rate to the service rate, is a crucial concept in queueing theory. It determines system stability—ensuring it remains less than one—and influences performance measures such as the average waiting time and probability of delay.

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