Queuing Theory

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Queuing Theory
Mathematical Programming ModelingENGC 6362
Lecture 9
Dr. Rifat Rustom
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Daily lives gas stations, stop signs,
supermarkets, restaurants, newsstands, and other
places. Transportation systems planes
circling an airport awaiting clearance from the
control tower, trucks waiting to load or unload
cargo, buses backed up waiting to enter a
terminal, cabs queuing up at airports and train
stations waiting for passenger ( or passengers
queuing up waiting for cabs), and ferries queuing
up waiting to off-load passengers and autos.
Banks and post offices. Factories jobs
queue up awaiting processing, orders need to be
filled, machines need repairs or need to be
loaded after a job, and employees wait to punch
the time-clock or to eat in the cafeteria.
Examples of waiting lines
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Most systems are characterized by highly variable
arrival and service rates. Consequently, even
though overall system capacity exceeds processing
requirements, lines tend to form from time to
time because of temporary system overload caused
by this variability. At other times, the reverse
is true variability in demand for service
results in idle servers or idle service
facilities because of a temporary absence of
customers.
Why Queues Occur?
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GOALS OF QUEUING SYSTEM DESIGN
Queuing models are predictive models of the
expected behavior of a system in which waiting
lines form.
A very common goal in queuing design is to
attempt to balance the cost of providing service
capacity with the cost of customers waiting for
service.
Total cost
Cost ()
Service capacity cost
Customer waiting cost
0
Minimum
System capacity
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CHARACTERISTICS OF A QUEUEING SYSTEM
KEY FEATURES
? A customer population, which is the collection
of all possible customers. ? An arrival process,
which is how the customers from this population
arrive. ? A queuing process, which consists of
(a) the way in which the customers wait for
service and (b) the queuing discipline, which is
how they are then selected for service. ? A
service process, which is the way and rate at
which the customers are served. ? Departure
processes.
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Components of a Queuing System
Servers
Waiting Customers
Customer Population
Arrival process
Queuing Process
Service Process
system
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Departure process of a Queuing System
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Case 1 Items leave the system completely after
being served, resulting in a one-step queuing
system. Example Customers at a bank wait in a
single line, are served by one of three tellers,
and, once served, leave the system.
Arrival
Depart
2
Waiting Customers
3
Tellers
(a)
Case 2 Items, once finished at a station, proceed
to some other work station to receive further
service, resulting in a network of queues.
Example Items are first processed at work
station A and then routed either to station B or
C. The finished items at both stations B and C
must then be processed at station D before
leaving the system.
B
Depart
Arrival
D
A
C
(b)
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The Queuing Process
1. Single Line Queuing System
Waiting Customers
2. Multiple-Line Queuing System
Servers
Waiting Customers
Servers
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Key Issues for Service Comparison of Single-and
Multiple-Channel Queuing System
Single channel, single phase
Exit
Single channel, multiple phase
Exit
Exit
Exit
Multiple channel, single phase
Exit
Exit
Exit
Single channel, multiple phase
Exit
Exit
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Queuing Discipline
First-in-First-Out (FIFO) The customers are
served in the order of their arrival in the line.
The customers in a bank and supermarket, for
example, are selected in this way. Last-in-First-O
ut (LIFO) The customer who has arrived most
recently is processed first. An example arises
in a production process where the items arrive at
a work station and are stacked one on top of the
other. The worker selects the item on top of the
stack for processing, which is the last one to
have arrived, for service. Priority Selection
Each arriving customer is given a priority and
selected for service accordingly. One example is
patients arriving at an emergency room of a
hospital. The more severe the case, the higher
the customers priority.
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A Poisson Distribution is usually used to
describe the variability in arrival rate
Arrival Rate
Relative frequency ()
0
1
2
3
4
5
6
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Number of arrivals
Interarrival Time
Relative frequency ()
If the arrival Rate Is Poisson, the Interarrival
Time is Negative Exponential
0
1
2
3
4
5
6
Time between arrivals (hours)
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Service Time
Negative Exponential Time Distribution
Relative frequency ()
0
Service time
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The Exponential Distribution
f(t) (1/?)e-? T
Where ? is the average number of arrivals per
unit of time T
Given an amount of time T, you can use this
density function to compute the probability that
the next customer arrives within T units of the
previous arrival, as follows
P(interarrival time lt T) 1- e-? t
Example if customers arrive at a bank at an
average rate of ? 20 per hour and if a customer
has just arrived, then the probability that the
next customer arrives within 10 minutes ( that is
, T 1/6 hour) is
P(interarrival time lt 1/6 hour ) 1 - e - 20
(1/6) 1 - e - 3.3333 1 -
0.036 0.964
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Poisson Distribution
e-? T (? T)k
P(number of arrivals in time T K )
K !
Where k! k(k - 1) . . . (2)(1). Example when ?
20 customers per hour and T 1/6 hour, the
probability of k 2 customers arriving within
the next 10 minutes is
e- (20)(1/6) ( 20/6)2
P(no. of arrivals in 10 min. 2 )
2 !
0.036 11.111

2

0.2 0
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System
Standard Notations
Queue or
Service
facility
Waiting line
1
2
Arriving Customers
Departing customers
X X X X X X
C
The main characteristics of parallel queues has
been universally standardized in the following
format, (a/b/c)(d/e/f) where the symbols a, b,
c, d, e, and f stand for basic elements of the
model as follows
a ? arrivals distribution
b ? service time (or departures) distribution
c ? number of parallel servers ( c 1, 2, . . .,
?)
d ? servers discipline (e.g., FCFS, LCFS, SIRO)
e ? maximum number allowed in system (in queue
in service)
f ? size of calling source
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The standard notation replaces the symbols a and
b for arrivals and departures
M ? Poisson (or Markovian) arrival or departure
distribution (or equivalently exponential
interarrival or/service-time distribution)
D ? constant or deterministic interarrival or
service time
Ek ? Erlangian or gamma distribution of
interarrival or service time distribution with
parameter K
GI ? General independent distribution of arrivals
(or interarrival time )
G ? General distribution of departures (or
service time )
To illustrate the notation, consider
(M/D/10)(GD/N/?)
Here we have Poisson arrivals, constant service
time, and 10 parallel servers in the facility.
The service discipline is general (GD) in the
sense that it could be FCFS, LCFS, SIRO, or
whatever procedure the servers may use to decide
on the order in which customers are chosen from
the queue to start service. Regardless of how
many customers arrive at the facility, the system
(queue service) can hold only a maximum of N
customers all others must seek service
elsewhere. Finally, the source generating the
arriving customers has an infinite capacity.
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Comparison of Single-and Multiple-Channel Queuing
system
Lq the average number waiting for service L
the avarege number in the system ( I.e., waiting
or being served) Po the probability of zero
units in the system p the system utilization
(percentage of time servers are occupied) Wqthe
average time customers must wait for service W
the average time customers spend in the system
(I.e., waiting for service and service time) M
the expected maximum number waiting for service
for a given level of confidence. One additional
measure is the total cost of the system, which is
generally based on the cost of customer waiting
time and the cost of server time.
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Comparison of Single-and Multiple-Channel Queuing
system
System
In the waiting line
Being served
µ
Average number
Lq
?
1
Average time
Wq
µ
where
? mean arrival rate µ mean service rate
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Basic Relationships
For converting performance measures from number
waiting to time waiting, and vice versa.
1. The average number being served
?
r

µ
2. The average number in the system
L Lq r
where
L average number in the system Lq average
number in line
3. The average time in line
Lq
Wq

?
4. The average time in the system, including
service
1
Wq
Ws


µ
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5. System utilization
?
?
sµ

Where s number of channels or servers
In order for a system to be feasible (i.e.,
underloaded), system utilization must be less
than 1.00.
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The owner of a car wash franchise intends to
construct another car wash in a suburban
location. Based on experience, the owner
estimates that the arrival rate for the proposed
facility will be 20 cars per hour and the service
rate will be 25 cars per hour. (Lets assume,
for the sake of illustration, that both arrival
and service rates are poisson. Note that a
poisson-distributed service rate is equivalent to
an exponential service time.) Service rate will
be variable because all cars are washed by hand
rather than by machine. Cars will be processed
one at time (hence, this is a single-channel, or
one-server, system). Determine the following a.
The average number of cars being washed. B. The
average number of cars in the system (I.e.,
either being washed or waiting to be washed), for
a case where the average number waiting in line
is 3.2. C. The average time in line (I.e., the
average time cars wait to get washed). D. The
average time cars spend in the system (I.e.,
waiting in line and being washed). E. The system
utilization.
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Solution Arrival rate, ?, 20 cars per hour
Service rate, µ, 25 cars per hour Number
of servers, s, 1 Lq 3.2
?
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a.
.80 cars being served
r
µ



25
L
Lq

b.

r
3.2 .8 4.0 cars
Lq
Wq
c.

?
3.2
.16 hour, which is .16 hour (60 minutes/hour


20 cars per hour
9.6 minutes
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1
Wq
Ws


d.
µ
1

1.6 Hour
Hour

.20 hour, which is .20 hour ( 60 minutes/hour) or
1.2 minutes

µ
?
20 cars per hour
e.
.80, or 80 percent
?
sµ



(1) (25 cars/per hour
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Basic Single-Channel Model
A single-channel model is appropriate when these
conditions exit
1.One server or channel. 2. A Poisson arrival
rate. 3. A negative exponential service time. 4.
Processing order is first-come, first-served. 5.
The calling population is infinite. 6. There is
no limit on queue length. The necessary formulas
for the single-server model are presented in Table
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Table 17-2 Formulas for Basic Single Server Model
Performance Measure
Formula
Formula Number
?
System utilization Average number in line Average
number in system Average time in line Average
time in system Probability of zero units in the
system Probability of n units in the
system probability the waiting line wont exceed
k units Average waiting time for an arrival not
served immediately
?
(17-6)

µ
?2
Lq
(17-7)
µ (µ- ?)

?
(17-8)
L Lq
µ
Lq
(17-9)
Wq

?
1
W
Wq
(17-10)

µ
?
Po 1 -
(17-11)
µ
n
?
(17-12)
Pn Po
µ
?
Pnltk 1 -
(17-13)
µ
1
Wa
(17-14)

µ - ?
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EAMPLE 2
The mean arrival rate of customers at a ticket
counter with one server is 3 per minute, and the
mean service rate is 4 customers per minute.
Calculate each of the performance measures listed
in Table 17-2. Assume that n 2 and K 5 .
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?
a.


.75, or 75 percent
4
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Lq
b.


2.25 customers.
4 (4 - 3)
3
L
2.25
c.

3.00 customers
4
2.25
Wq
d.
.75 minutes


3
1
W
.75
e.
1.00 minute

4
3
Po
1 -
f.
.25.
This means that the probability is 25 percent
that an

4
arriving unit will not have to wait for service.
Hence, the probability that an arrival will have
to wait for service is 75 percent.
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2
3
Pn2
g.

.25

.1406
4
4 1
3
Pnlt4
h.

1 -

.7627
4
1
Wa
i.
1.0 minutes


4 - 3