Supplement E: Waiting Line Models

Supplement E: Waiting Line Models

Why They Form

·  When demand temporarily exceeds capacity

·  Variable demand and service rates

Use of waiting line theory

·  Types of service or manufacturing situations

·  OM decision areas

Structure of Waiting Line Problems

Suppose you are in charge of designing a drive-in window system for a branch of your bank. How would you describe this service system as a waiting line problem?

1. Customer population

·  Finite or infinite

·  Patient or impatient

Ø  Balking (not to enter)

Ø  Reneging (leave before being served)

2. Waiting line(s)

·  Single line

·  Multiple lines

3. Service facility

·  Single-channel, single-phase

·  Single-channel, multiple-phase

·  Multiple-channel, single-phase

·  Multiple-channel, multiple-phase

·  Mixed

4. Priority rule

·  FCFS

·  Other commonly used rules

·  Preemptive discipline

Operating Characteristice

1.  Line Length

2.  Number of customers in system

3.  Waiting time in line

4.  Total time in system

5.  Service facility utilization

Probability Distributions

1.  Arrival distribution

·  Arrivals per unit of time (Poisson distribution)

·  Interarrival time (exponential distribution)

where l is average # of arrivals per period (i.e., arrival rate)

2. Service-time distribution

·  Time to complete service at a service facility (exponential distribution)

·  Assumptions

Single-server model

a. Assumptions and formulas

Assumptions / Formulas
Number of servers: / 1 /
Number of phases: / 1 /
Input source: / infinite; no balking or reneging /
Arrival distribution: / Poisson; mean arrival rate = l /
Service distribution: / exponential; mean service time = 1/m /
Waiting line: / single line; unlimited length /
Priority rule: / first-come, first served

Application 1

Customers arrive at a checkout counter at an average 20 per hour, according to a Poisson distribution. They are served at an average rate of 25 per hour, with exponential service times. Use the single-server model to estimate the operating characteristic of this system.

r =

L =

Lq =

W =

Wq =

Application 2

In the checkout counter example, what service rate is required to have customers average only 10 minutes in the system?

Multiple-server model

a. Assumptions and formulas

Assumptions / Formulas
Number of servers: / s /
Number of phases: / 1 /
Input source:
Arrival distribution: / infinite; patient,
no balking or reneging
Poisson;
mean arrival rate = l /
Service distribution: / exponential; mean service time = 1/m /
Waiting line: / single line; unlimited length /
Priority rule: / first-come, first served /

Application 3:

Suppose the manager of the checkout system decides to add another counter. The arrival rate is still 20 customers per hour, but now each checkout counter will be designed to service customers at the rate of 12.5 per hour. What is the waiting time in line of the new system?

Note that even though the average service rate is the same in the two systems, the average waiting time is less in the multiple-server arrangement.

Supplement E: Waiting Line Models LNE-3