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
C-2
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)
C-3
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
C-4
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
 S1 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
C-10
(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 
C-11
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
C-17