# Economics of the Waiting Line Problem

A central problem in many service settings is the management of waiting time.

Reducing waiting time costs money.

When people waiting are employees, it is easy to value their time.

When people waiting are customers, it is more difficult to value their time.

Lost sales is one (low) value.

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The Practical View of Waiting Lines

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More on Waiting Lines

One important variable is the number of arrivals over the hours that the service system is open.

Customers demand varying amounts of service, often exceeding normal capacity.

We can control arrivals.

Short lines

Specific hours for specific customers

Specials

We can affect service time by using faster or slower servers.

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The Queuing System

Source population and the way customers arrive at the system

The servicing system

The condition of the customers exiting the system

Do they go back to source population or not?

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Components of the Queuing System Visually

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Customers come in

Customers are served

Customers leave

Customer Arrivals

Finite population: limited-size customer pool that will use the service and, at times, form a line

When a customer leaves his/her position as a member of the population, the size of the user group is reduced by one.

Infinite population: population large enough so that the population size caused by subtractions or additions to the population does not significantly affect the system probabilities

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Distribution of Arrivals

Arrival rate: the number of units arriving per period

Constant arrival distribution: periodic, with exactly the same time between successive arrivals

Variable (random) arrival distributions: arrival probabilities described statistically

Exponential distribution

Poisson distribution

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Customer Arrivals in Queues

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Other Arrival Characteristics

Arrival patterns

Size of arrival units

Degree of patience

Balking

Reneging

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The Queuing System

Length

Infinite potential length

Limited line capacity

Number of lines

Queue discipline: a priority rule or set of rules for determining the order of service to customers in a waiting line

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Service Time Distribution

Constant

Service provided by automation

Variable

Service provided by humans

Described using exponential distribution

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Line Structure

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Exiting the Queuing System

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Properties of Some Specific Waiting Line Models

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Notation for Equations

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Equations for Solving Three Model Problems

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Example 10.1: Customers in Line

Western National Bank is considering opening a drive-through window for customer service. Management estimates that customers will arrive at the rate of 15 per hour. The teller who will staff the window can service customers at the rate of one every three minutes.

Part 1 Assuming Poisson arrivals and exponential service, find

Utilization of the teller

Average number in line

Average number in system

Average waiting time in line

Average waiting time in system, including service

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Example 10.1: Solution

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Computer Simulation of Waiting Lines

Some waiting line problems are very complex.

Assumed waiting lines are independent.

When a services is becomes the input to the next, we can no longer use the simple formulas.

This is also true for any problem where conditions do not meet the requirements of the equations.

Here, computer simulation must be used.

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Simulating Waiting Lines

Waiting lines that occur in series and parallel cannot be solved mathematically.

Assembly lines

Work centers

These waiting lines are easily simulated on a computer.

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