Waiting Line

Topics: Probability theory, Normal distribution, Poisson distribution Pages: 50 (3562 words) Published: June 1, 2014
Ch 12. Waiting Line Models
Contents
1. Structure of Waiting Line System
2. Single-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times
3. Multiple-Channel Waiting Line Model with Poisson Arrivals and Exponential Service Times
4. Economic Analysis of Waiting Lines
5. Other Waiting Line Models
6. Single-Channel Waiting Line Model with Poisson Arrivals and Arbitrary Service Times
7. Multiple-Channel Model with Poisson Arrivals, Arbitrary Service Times and No Waiting Line
8. Waiting Line Model with Finite Calling Population

9. Estimations of Arrival Process and Service Time Distribution 1

권치명

Waiting Line Models
 Waiting line or Queue



Model is developed to help manager make to better decision for the operation of waiting line.
Erlang (a Danish Telephone engineer) began a study of
congestion and waiting times in the completion of telephone
calls.

 Operating Characteristic (performance Measure) for a
waiting Line Model





Probability that no units are in the system
Probability that an arriving unit has to wait for service
Average Number of units in waiting line or system
Average Time a unit spends in waiting line or system

 Make a decision that balance desirable service level
against the cost of providing the service

2

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1. Structure of Waiting Line System
 Single-channel Waiting Line System
Service Rule

Server

Costumer
Arrives

Customer
Leaves
Waiting Line

Service System

 Elements for Waiting Line System





Population of arrivals and their arrival process
Capacity of waiting Line
Service discipline and service facility structure
Service process
3

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1.1 Population of Arrivals
 Infinite Population of arrival





Population of arrival to system is finite in most cases.
Customers to watch the movie ‘Abata’ may live in the Busan metropolitan area.
For the convenience of analysis, we assume that it is an infinite population.

 Finite Population of arrival


When the size of population of arrivals to system is small and the probability of costumer’s arrival is dependent of its size, we assume the population of arrivals is finite.



A factory operates 10 machines. An arrival process of the breakdowned machines to repair-shop is dependent on the number of machines break-downed.

4

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1.2 Arrival Process
 Arrival process of customers to system
 The arrivals occur randomly and independently.
 The Inter-arrival time is described either by historical distribution or theoretical distribution.
 Mean arrival rate is the number of customers per unit
time.






When the mean arrival rate is 3 persons/hour, the mean interarrival time is 20min/person. The mean arrival rate is a reciprocal of the mean inter-arrival time.

Generally in waiting line theory, the Poisson arrival
process provides a good description of the arrival pattern.

5

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1.3 Poisson Arrival Process
Assumptions for Poisson arrival process
(1) During an arbitrary time period t, the probability that
customers arrive to system is proportional to time period t. (2) For an infinitely small period, the probability that
customers more than 1 arrive to system is 0.
(3) The number of costumers that arrive to system for a
time period is independent of that for mutually not
overlapped period.

Only Poisson distribution satisfies the above 3
assumptions.

6

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1.4 Poisson distribution
Poisson distribution







Mean arrival rate = 
Mean number of arrival
for an interval (0, t) or (s, s+ t) = m  t
Number of arrivals for (0, t) or (s, s +t) = x
e = 2.71828
The number of arrivals for (0, t) follows Poisson
distribution.

( t ) e
p( x) 
x!
x

7

 t

, x  0,1,2,3, 

권치명

Example 1
Burger Dome’s costumer arrival data
 Mean arrival rate =   0.75






Mean number of arrivals
for an interval (0, 1) = m ...
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