2. Introduction
Models to help managers understand and make decisions on the
operation of waiting lines.
Also known as a queue -- waiting line models are based on queueing
theory.
We are interested in the operating characteristics (a.k.a performance
characteristics) of waiting lines, which include the following:
1. The probability that no units are in the system.
2. The average number of units in the waiting line.
3. The average number of units in the system (number of units in
the waiting line + number being served)
4. The average time a unit spends in the waiting line.
5. The average time a unit spends in the system (waiting time +
service time)
6. The probability that an arriving unit has to wait for service.
3. General Waiting Line System Characteristics
Single Channel Waiting Line - Each customer entering an establishment passes through
one channel -- for example, one line and one order taking and filling station.
For most waiting line situations customer arrivals occur randomly and independently.
Quantitative analysts have found the Poisson distribution provides a good distribution
of the arrival pattern.
The probability function is as follows:
x = the number of arrivals in the time period
λ = the mean number of arrivals per time period (arrival rate)
e = 2.71828
While we can use this, in practice data should be recorded over a period of several days
or weeks and compared to the Poisson distribution.
4. General Waiting Line System Characteristics
Distribution of Service Times - the time a customer spends at a
service facility once a service has started.
Service time has been found to have an exponential probability
distribution.
µ = the mean number of units that can be served per time period (service rate)
e = 2.71828
Again, while we can use this, in practice data should be recorded over a period of
several days or weeks and compared to the Exponential distribution.
Additional Terms:
FCFS - First come first served -- this is the model we use in this section.
Transient Period - the beginning or start up period.
Steady-State Operation - normal state of operations.
5. Operating Characteristics - Single Channel Waiting Line Model with
Poisson Arrivals and Exponential Service Times
6. Operating Characteristics - Multiple Channel Waiting Line Model
with Poisson Arrivals and Exponential Service Times
These formulas are only applicable if:
1. The arrivals follow a Poisson
probability distribution
2. The service time for each channel
follows an exponential probability
distribution.
λ = the arrival rate
µ = service rate
k = number of channels
