SMU
EMIS 5300/7300
NTU
SY-521-N
Systems Analysis Methods
Dr. Jerrell T. Stracener,
SAE Fellow
Queuing Modeling and Analysis
updated 11.16.01
1
Queuing Models
• Several models exist, depending on the
structure of the system, the nature of arrivals,
the service policies, and the behavior of the
customers in the queue.
• These queuing situations are commonly
designated X/Y/Z where X indicates the arrival
process, Y indicates the service process, and
Z the number of servers.
2
Queuing Models
• Some queuing situations are:
1. Single server (1), Poisson arrivals (M),
exponential service (M), called M/M/1.
2. Single server, Poisson arrival, exponential
service, with finite (limited) queue length:
M/M/1 finite queue.
3. M/M/1 finite source (a finite calling population).
4. M/arbitrary/1 (arbitrary service time distribution,
but mean and standard deviation are known).
5. M/M/K (multiple servers: K).
6. M/M/K finite queue.
7. M/M/K finite source.
8. M/constant/K (constant service times).
3
Solution Approaches
There are two basic approaches to the solution
of queuing problems: analytical and simulation.
• The analytic approach - The measures of
performance are determined through the use of
formulas. Unfortunately, many queuing situations
are so complex that the analytic approach is
completely impractical or even impossible.
• Simulation - For those situations in which the
analytic approach is unsuitable, the procedure of
simulation can be used.
4
Information Flow in Waiting Line Models
• It is helpful to use some measures of performance
when evaluating service alternatives, particularly
when a cost approach is planned.
• A solution to a queuing problem means computing
certain measures of performance
• These measures are computed from three input
variables:
l, the mean arrival rate
m, the mean service rate
, the number of servers
5
Information Flows
m
l

W = Average waiting time, per customer
in the system
Wq = Average waiting time, per customer
in the queue
L = Average number of customers in the
system
Lq = Average number of customers in the
queue
P(0) = Probability of the system being idle
Pw = Probability of the system being busy
P(t > T) = Probability of waiting longer than
time T
P(n) = Probability of having exactly n customers
in the system
P(n > N), P(n < N) - Probability of finding more
than, or less than, N customers in the system
6
Deterministic Queuing Systems
The simplest and the rarest of all waiting line
situations involves constant arrival rates and
constant service times. Three cases can be
distinguished:
1. Arrival rate equals service rate. Assume that
people arrive every 10 minutes, to a single server,
where the service takes exactly 10 minutes. Then
the server will be utilized continuously (100%
utilization), and there will be no waiting line.
7
Deterministic Queuing Systems
2. Arrival rate larger than service rate. Assume
that there are six arrivals per hour (one every 10
minutes) and the service rate is only five per hour
(12 minutes each). Therefore, one arrival cannot
be served each hour, and a waiting line will build
up (at a rate of one per hour). Such a waiting line
will grow and grow as time passes and is termed
explosive.
8
Deterministic Queuing Systems
3. Arrival rate smaller than service rate. Assume
that there are again six arrivals per hour but the
service capacity is eight per hour. In this case the
facility will be utilized only 6/8 = 75% of the time.
There will never be a waiting line (if the arrivals
come at equal intervals).
9
The Basic Poisson-Exponential Model (M/M/1)
The classical and probably best known of all
waiting line models is the Poisson-exponential
single server model. It exhibits the following
characteristics.
Arrival rate - The arrival rate is assumed to be
random and is described by Poisson distribution.
The average arrival rate is designated by the Greek
letter l.
Service time - The service time is assumed to follow
the negative exponential distribution. The average
service rate is designated by the Greek letter m, and
the average service time by 1/m.
10
The Major Ground Rules for the Operation of a
Single Server System
• Infinite source of population
• First-come, first-served treatment
• The ratio l/m is smaller than 1. This ratio is
designated by the Greek letter . The ratio is a
measure of the utilization of the system. If the
utilization factor is equal to or larger than 1, the
waiting line will increase without bound (will be
explosive), a situation which is unacceptable to
management.
11
The Major Ground Rules for the Operation of a
Single Server System
• Steady state (equilibrium) exists. A system is
in a ‘transient state’ when its measures of
performance are still dependent on the initial
conditions. However, our interest is in the ‘long
run’ behavior of the system, commonly known as
steady state. A steady state condition occurs when
the system becomes independent of time.
• Unlimited queuing space exists.
12
Managerial Use of the Measures of Performance
Some of these measures can be used in a cost
analysis, while others are used to aid in
determining service level policies. For example:
a. A fast-food restaurant wants to design its
service facility such that a customer will not wait,
on the average, more than two minutes (i.e.,
Wq  2 minutes) before being served.
b. A telephone company desires that the
probability of any customer being without
telephone service more than two days be 3%
(i.e., P(t > 2 days) = 0.03
13
Managerial Use of the Measures of Performance
c. A bank’s policy is that the number of customers
at its drive-in facility will exceed 10 only 5% of
the time (i.e., P(n > 10) = 0.05.
d. A city information service should be busy at
least 60% of the day (i.e., Pw > 0.6).
14
Example - The Toolroom problem
The J.C. Nickel Company toolroom is staffed by one
clerk who can serve 12 production employees, on
the average, each hour. The production employees
arrive at the toolroom every six minutes, on the
average. Find the measures of performance.
15
Example - The Toolroom problem solution
It is necessary first to change the time dimensions
of l and m to a common denominator. l is not given
in minutes, m in hours. We will use hours as the
common denominator.
1. Average waiting time in the system (toolroom)
1
1
W

 0.5
m  l 12  10
hours, per employee
2. The average waiting time in the line.
10
10
Wq 

 0.417
hours,
m m  l  1212  10
per employee 16
Example - The Toolroom problem solution
3. The average number of employees in the
toolroom area
l
10
L

5
m  l 12  10
employees
4. The average number of employees in the line.
l
100
Lq 

 4.17
m m  l  1212  10
2
employees
17
Example - The Toolroom problem solution
5. The probability that the toolroom clerk will be
idle.
l
10
P0  1   1   0.167
m
12
6. The probability of finding the system busy.
 l  10
Pw      0.833
 m  12
18
Example - The Toolroom problem solution
7. The chance of waiting longer than 1/2 hour in
the system. That is T = 1/2.
Pt  T   e
1012 1/ 2 
1
  0.368
e
8. The probability of finding four employees in
the system, n = 4.
λ
P4   
μ
n
 λ   10 
1     
 μ   12 
4
2
   0.0804
 12 
19
Example - The Toolroom problem solution
9. The probability of finding more than three
employees in the system.
l
Pn  3   
m
N 1
4
 10 
    0.488
 12 
20