Wednesday, February 27, 2008

Introduction to Waiting Lines


We come in contact with waiting line systems, or queuing systems, everywhere and everyday. May it be waiting in line for your morning coffee, opening up your email account to see your list of new messages, or stopping at a red light in a traffic intersection, you are participating in a waiting line system.

As a business, waiting line systems are especially important to the operations management of the organization.
Examples of the costs and benefits of these systems are:


  • lost customers from either balking (customer refuses to enter the waiting line because it is too long) or reneging (subject leaves the waiting line after entering it because it is too long
  • lower customer satisfaction that can cause damage to brand image


  • psychological boost to build anticipation and excitement
  • a value or price-tag attached for the rarity of the product or service. Such example is the line-up for the release of iPhone shown in the picture on the right.

Structure of waiting lines:

Each specific situation will be different, but waiting line systems are essentially composed of four major elements:

  1. An input of customers or items to make a customer population. This population can be finite or infinite

  2. A waiting line system of customers or items

  3. Workstations or operations that perform one or more activities

  4. A priority rule that selects the next customer or item on which the activities perform

Layout and Arrangements:

Management has many decisions to make as to how operations can optimize the waiting line system, one of which is layout and arrangements of the system that can be tinkered to best fit the overall process. Some examples are:

  1. Number of lines: single lines are most commonly used for workstations that provide uniform services and will prevent jockeying (customer switches lines for shortest estimated wait times); multiple lines are used when specialty services or activities are offered, such as express-lanes or cash-only lanes
  2. Number of phases: depending on customer volume and nature of the process, the number of steps performed in different workstations, or phases, can be varied along with differing numbers of channels, which is the number of lines that a potential customer or item can arrive from to a workstation

Many companies have creatively utilized their waiting areas to actually provide a positive experience for their customers. These areas are commonly near the lobby and reception desk and are the first areas that the customer come in contact with. Companies often use this opportunity to add in extra effort to accessorize and decorate the area to create an image for the customer. Such examples are waiting lounge areas in restaurants as displayed on the right.

Priority Rules:

Priority Rules make up the final element of the queuing system, which determines which customer or item is to be served next. Some examples are:

  • First-come, first served (FCFS)

  • Earliest promised due date (EDD)

  • Shortest expected processing time (SPT)
  • Pre-emptive discipline, which allows a customer of higher priority to interrupt the service of another customer. An example is the waiting line system in a hospital emergency room where the most life-threatening injuries receive treatment first

Waiting Line Models using Poisson Distributions:

To begin a systematic analysis of waiting lines, one beings with calculating the probability and distribution of arrivals. The variability of customer arrivals often can be described by a Poisson distribution, which specifies the probability that n customers will arrive in T time periods:

P(n) = (((lambda * T)^n) / (n!))* e^-(lambda*T)


P(n) = Probability of n arrivals in T time periods

lambda = Average number of customer arrivals per period

e = 2.7183

lambda*T = mean and variance of Poisson distribution

Single and Multi-Workstation Models:

With more in-depth calculations, one can utilize such probability data to determine the following data using either a single or multi-workstation model. Description on these calculations are far beyond the scope of this post, but please visit the following document for the detailed analysis.

  1. Average capacity utilization of the system

  2. Probability that n customers are in the system

  3. Probability that zero customers are in the system

  4. Average number of customers in the system

  5. Average number of customers in the waiting lines
  6. Average time spent in the system, including waiting and operations time

  7. Average waiting time in line