Module 8A Queuing Model Summary
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Transcript Module 8A Queuing Model Summary
Queuing Model Summary
C-1
Assumptions of the Basic Simple
Queuing Model
Arrivals are served on a first-come, first-served
basis (FCFS)
Arrivals are independent of preceding arrivals
Arrival rates are described by the Poisson
probability distribution, and customers come from
a very large population
Service times vary from one customer to another,
and are independent of each other; the average
service time is known
Service times are described by the negative
exponential probability distribution
The service rate is greater than the arrival rate
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Types of Queuing Models
(A/B/C notation)
A: probability distribution of time between arrivals
B: probability distribution of service times
C: number of parallel servers
M = exponential distribution of times (or equivalent Poisson
distribution of rates)
D = deterministic or constant time
G = general distribution with a mean and variance (e.g., normal,
uniform, or any empirical distribution)
Ek = Erlang distribution with shape parameter k (if k =1, Erlang
equivalent to M; if k = ∞, Erlang equivalent to D)
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Types of Queuing Models
(A/B/C notation)
Simple (M/M/1)
Example: Information booth at mall, line at Starbucks
Multi-channel (M/M/S)
Example: Airline ticket counter, tellers at bank
Constant Service (M/D/1)
Example: Automated car wash
Limited Population
Example: Department with only 7 copiers to service
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Simple (M/M/1) Model
Characteristics
Type: Single-channel, single-phase system
Input source: Infinite; no balks, no reneging
Arrival distribution: Poisson
Queue: Unlimited; single line
Queue discipline: FIFO (FCFS)
Service distribution: Negative exponential
Relationship: Independent service & arrival
Service rate > arrival rate
C-5
Simple (M/M/1) Model Equations
Ls =
Average number of units
in the system
-
Ws =
Average time in the
system
1
-
2
Lq =
Average number of units in
the queue
( - )
( - )
=
Wq =
Average time waiting in
the queue
System utilization
C-6
Simple (M/M/1) Probability
Equations
Probability of 0 units in system, i.e., system idle:
P = 1- = 10
Probability of more than k units in system:
P =
n>k
()
k+1
Where n is the number of units in the system
C-7
Multichannel (M/M/S) Model
Characteristics
Type: Multichannel system
Input source: Infinite; no balks, no reneging
Arrival distribution: Poisson
Queue: Unlimited; multiple lines
Queue discipline: FIFO (FCFS)
Service distribution: Negative exponential
Relationship: Independent service & arrival
Individual server service rates > arrival rate
C-8
(M/M/S) Equations
Probability of zero
people or units in the
system:
P0
1
S1 1 λ n 1 λ S Sμ
n 0 n! μ S! μ Sμ λ
S
Average number of
people or units in the
system:
Ls
Average time a unit
spends in the system:
1
Ws
P
2 0
S 1!S
P0
S 1!S
S
C-9
P0 = Probability of 0 Units in MultipleChannel System
(needed for other calculations)
P0
1
S 1 1 n 1 S S
n 0 n! S! S
n! = 1 x 2 x 3 x 4 x……..x (n-1) x n
n0 = 1; 0! = 1
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(M/M/S) Equations
Lq L s
Average number of
people or units
waiting for service:
Average time a
person or unit spends
in the queue
Wq W s
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1
Constant Service Rate (M/D/1)
Model Characteristics
Type: Single-channel, single-phase system
Input source: Infinite; no balks, no reneging
Arrival distribution: Poisson
Queue: Unlimited; single line
Queue discipline: FIFO (FCFS)
Service distribution: Constant
Relationship: Independent service & arrival
Service rate > arrival rate
C-12
(M/D/1) Equations
Average number of people
or units waiting for service:
Lq
Average time a person or
unit spends in the queue
Wq
Average number of people or
units in the system:
Ls L q
Average time a unit spends in
the system:
Ws Wq
C-13
Limited Population Model
Characteristics
Type: Single-channel, single-phase system
Input source: Limited; no balks, no reneging
Arrival distribution: Poisson
Queue: Limited; single line
Queue discipline: FIFO (FCFS)
Service distribution: Negative exponential
Relationship: Independent service & arrival
Service rate > arrival rate
C-14
Single-Channel, Single-Phase
Manual Car Wash Example
Arrival rate = 7.5 cars per hour
Service rate = an average of 10 cars per hour
Utilization = / = 75%
C-15
Single-Channel, Single-Phase
Automated Car Wash Example
Arrival rate = 7.5 cars per hour
Service rate = a constant rate of 10 cars per hour
Utilization = / = 75%
C-16
Comparisons
Manual wash,
single server
Automated wash,
single server
Manual wash, two
servers
Cars
waiting
2.25
1.125
0.1227
Cars in
system
3
1.875
1.517
Time
waiting
18 minutes
9 minutes
1 minute
Time in
System
24 minutes
15 minutes
7 minutes
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