Fall 2010
Part 5: Queueing Models
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Agenda
1. The Purpose
2. Characteristics of Queues Systems
3. Queueing Notations
4. Long-Run Measures
5. Steady-State Behavior of Infinite-Population Markovian
Models
6. Steady-State Behavior of Finite-Population Markovian
Models
Fall 2010
7. Final Summary
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1. The Purpose (1): Queueing
System Examples
Fall 2010
Table 6.1
Exam ples of Queueing Sys tem s
System
Recep tion desk
Rep air facility
Garage
Tool crib
Hosp ital
Warehouse
Airp ort
Production line
Warehouse
Road network
Grocery
Laundry
Job shop
Lumbery ard
Sawmill
Comp uter
Telep hone
Ticket office
M ass transit
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Customers
Peop le
M achines
Trucks
M echanics
Patients
Pallets
Airp lanes
Cases
Orders
Cars
Shop p ers
Dirty linen
Jobs
Trucks
Logs
Jobs
Calls
Football fans
Riders
Server(s)
Recep tionist
Rep airp erson
M echanic
Tool-crib clerk
Nurses
Fork-lift Truck
Runway
Case-p acker
Order-p icker
Traffic light
Checkout station
Washing machines/dry ers
M achines/workers
Overhead crane
Saws
CPU, disk, CDs
Exchange
Clerk
Buses, trains
2
1. The Purpose (2)
Simulation
is often used in the analysis of
queueing models.
Queueing
models provide the analyst with a
powerful tool for designing and evaluating the
performance of queueing systems.
Discuss some well-known models (NOT
development of queueing theories)
Fall 2010

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2. Characteristics of Queueing
Systems (1): Key Elements
Key elements of queueing systems:
• Customer: refers to anything that arrives at a facility and
requires service, e.g., people, machines, trucks, emails.
• Server: refers to any resource that provides the requested service,
e.g., repair persons, retrieval machines, runways at airport.
Waiting Line
Server(s)
Customer Arrivals
Fall 2010
Calling Population
Arrival Process
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System Capacity
Queue Behavior
Queue Discipline
Service Times
Service Mechanisms
4
2. Characteristics of Queueing
Systems (2): Calling Population
Calling population: the population of potential customers, may be
assumed to be finite or infinite.
• Finite population model: if arrival rate depends on the number of
customers being served and waiting, e.g., model of one corporate
jet, if it is being repaired, the repair arrival rate becomes zero.
Fall 2010
• Infinite population model: if arrival rate is not affected by the
number of customers being served and waiting, e.g., systems with
large population of potential customers.
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2. Characteristics of Queueing
Systems (3): System Capacity
System Capacity: a limit on the number of customers that may
be in the waiting line or system.
• Limited capacity, e.g., an automatic car wash only has room for 10
cars to wait in line to enter the mechanism.
Fall 2010
• Unlimited capacity, e.g., concert ticket sales with no limit on the
number of people allowed to wait to purchase tickets.
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2. Characteristics of Queueing
Systems (4): Arrival Process (1)
Fall 2010
For infinite-population models:
• Usually characterized in terms of inter-arrival times of successive
customers. Arrivals may occur at random or scheduled times
• Random arrivals: inter-arrival times usually characterized by a
probability distribution.
– Most important model: Poisson arrival process (with rate l),
where An represents the inter-arrival time between customer
n-1 and customer n, and is exponentially distributed (with
mean 1/l).
• Scheduled arrivals: inter-arrival times can be constant or
constant plus or minus a small random amount (jitter) to
represent early or late arrivals.
– e.g., patients to a physician or scheduled airline flight arrivals
to an airport.
• At least one customer is assumed to always be present, so the
server is never idle, e.g., sufficient raw material for a machine.
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2. Characteristics of Queueing
Systems (4): Arrival Process (2)
For finite-population models:
• Define customer as “pending” when the customer is outside the
queueing system, e.g., machine-repair problem: a machine is
“pending” when it is operating, it becomes “not pending” the
instant it demands service form the repairman.
• Define “runtime” of a customer as the length of time from
departure from the queueing system until that customer’s next
arrival to the queue, e.g., machine-repair problem, machines are
customers and a runtime is time to failure.
Fall 2010
• Let A1(i), A2(i), … be the successive runtimes of customer i, and
S1(i), S2(i) be the corresponding successive system times: that is
Sn(i) is the total time spent in the system by customer i during the
nth visit.
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2. Characteristics of Queueing
Systems (4): Arrival Process (3)
Finite Population Models
Fall 2010
• The total arrival process is the superposition of the arrival
times of all customers.
• One important application of finite models is the machinerepair problem. Machines are the customers and runtime is
time to failure. When a machine fails, it “arrives” at the
queueing system and remains there until it is served. Time to
failure is chracterized by exponential, Weibull and Gamma
distributions.
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2. Characteristics of Queueing Systems (5):
Queue Behavior and Queue Discipline
Queue behavior: refers to the actions of customers while in a
queue waiting for service to begin, for example:
• Balk: leave when they see that the line is too long,
• Renege: leave after being in the line when its moving too
slowly,
• Jockey: move from one line to a shorter line.
Fall 2010
Queue discipline: refers to the logical ordering of customers in
a queue that determines which customer is chosen for
service when a server becomes free, for example:
•
•
•
•
•
First-in-first-out (FIFO)
Last-in-first-out (LIFO)
Service in random order (SIRO)
Shortest processing time first (SPT)
Service according to priority (PR). (e.g., type, class, priority)
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2. Characteristics of Queueing Systems (6):
Service Times and Service Mechanism (1)
Service times of successive arrivals are denoted by S1, S2, S3, ……
• May be constant or random.
• {S1, S2, S3, …} is usually characterized as a sequence of
independent and identically distributed random variables,
e.g., exponential, Weibull, gamma, lognormal, and truncated
normal distribution.
• Sometimes, services are identically distributed for all
customers of a given type or class or priority, where as
customers of different types might have completely different
service-time distributions
Fall 2010
• In some systems, service times depend upon the time of the
day or upon the length of waiting line (e.g., servers might
work faster than usual if waiting times are long, effectively
reducing service times)
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2. Characteristics of Queueing Systems (6):
Service Times and Service Mechanism (2)
A queueing system consists of a number of service centers and
interconnected queues.
• Each service center consists of some number of servers, c, working
in parallel
• upon getting to the head of the line, a customer takes the 1st
available server.
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• Parallel service mechanisms are either single server (c=1),
multiple server (1<c<), or unlimited servers (c=)
– A self service facility is usually characterized by an unlimited
number of servers
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2. Characteristics of Queueing Systems (6):
Service Times and Service Mechanism (3)
Example: consider a discount warehouse where customers may:
Fall 2010
• Serve themselves before paying at the cashier:
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2. Characteristics of Queueing Systems (6):
Service Times and Service Mechanism (4)
• Wait for one of the three clerks:
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• Batch service (a server serving several customers simultaneously,
e.g., small orders), or customer requires several servers
simultaneously (e.g., big order or heavy item).
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3. Queueing Notation (1)
Recognizing the diversity of queueing systems, Kendall proposed a
notational system in 1953 that has been widely adopted.
A notation system for parallel server queues: A/B/c/N/K
• A represents the inter-arrival-time distribution,
• B represents the service-time distribution,
• c represents the number of parallel servers,
N represents the system capacity,
• K represents the size of the calling population.
Fall 2010
Common symbols for A and B include M (exponential), D (constant
or deterministic), Ek (for Erlang order k), PH (Phase-type), H
(hyper-exponential), G (arbitrary or general), and GI (general
independent).
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3. Queueing Notation (2)
Examples:
• M/M/1 (also M/M/1//) indicates a single-server that has
unlimited queue capacity and an infinite population model. The
interarrivals and service times are exponentially distributed.
– When N and K are infinite, they are often dropped from the
notation
Fall 2010
• G/G/1/5/5 represents a queueing system with general (or
arbitrary) inter-arrival and service distribution with single
server, with a queue capacity of 5 and finite population model of
size 5
– General models are used to solve the queue system with no
particular distribution in mind
– Very useful as the final results can be obtained by plugging in
the values of specific distributions
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Fall 2010
3. Queueing Notation (3)
Primary performance measures of queueing systems:
 Pn: steady-state probability of having n customers in system,
 Pn(t):
probability of n customers in system at time t,
 l: arrival rate,
 le: effective arrival rate,
 m: service rate of one server,
 r: server utilization,
 An: inter-arrival time between customers n-1 and n
 Sn: service time of the nth arriving customer,
 Wn: total time spent in system by the nth arriving customer,
 WnQ: total time spent in the waiting line by customer n,
 L(t): the number of customers in system at time t,
 LQ(t):
the number of customers in queue at time t,
 L: long-run time-average number of customers in system,
 LQ: long-run time-average number of customers in queue,
 w: long-run average time spent in system per customer,
 wQ: long-run average time spent in queue per customer.
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4. Long-Run Measures (1): Time-Average
Number in System L (1)
Consider a queueing system over a period of time T,
• Let Ti denote the total time during [0,T] in which the system contained
exactly i customers, the time-weighted-average number in a system is
defined by:
1
Lˆ 
T



iTi 
i 0
T 
i i 
T
i 0 

• Consider the total area under the function is L(t), then, time-integratedaverage is given by:
Fall 2010
1
Lˆ 
T


1
iTi 
T
i 0
T
 L(t)dt
0
These two expressions are always equal for any queueing system regardless
of the number of servers, the queue discipline or any other circumstances
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4. Long-Run Measures (1): Time-Average
Number in System L (2)
Many queueing systems exhibit long-run stability in terms of their
average performance. For such systems, as time T gets large, the observed
time-average number in the system Lˆ approaches a limiting value L,
called long-run time-average.
The long-run time-average # in system L, with probability 1:
ˆL  1 T L(t )dt  L as T  
T 0
Fall 2010
The estimator Lˆ is said to be strongly consistent for L. If simulation run
length T is sufficiently long, the estimator Lˆ becomes arbitrarily close to
L. Unfortunately for T < , it depends on the initial condition at t=0.
(Reason to do multiple simulation runs!!)
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4. Long-Run Measures (2):
Time-Average Number in Queue LQ
(1)
Similarly, if LQ(t) denotes the number of customers waiting in line and TiQ
denotes the total time during [0,T] in which exactly i customers are
waiting in line. Then,
The time-weighted-average number in queue is:
1
LˆQ 
T


1
iTi 
T
i 0
Q
T
L
0
Q (t )dt
 LQ as T  
Here, LˆQ is the observed time-average number of customers in waiting
line from time 0 to T.
Fall 2010
LQ is the long-run time-average number waiting in line.
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4. Long-Run Measures (2):
Time-Average Number in Queue LQ
(2)
G/G/1/N/K example: consider the results from the queueing system
(N > 4, K > 3).
Fall 2010
Lˆ  [0(3)  1(12)  2(4)  3(1)] / 20
 23 / 20  1.15 cusomters
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if L( t) 0
0,
LQ (t )  
 L(t )  1, if L( t) 1
0(15)  1(4)  2(1)
LˆQ 
 0.3 customers
20
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4. Long-Run Measures (3): Average Time Spent
in System Per Customer w (1)
If we simulate the queueing system for a period, say T and then
record the time each customer spends in the system during [0,T],
say W1, W2, …., WN where N is the number of arrivals in [0,T].
The average time spent in system per customer, called the average
system time, is:
1
ˆ
w
N
N
W
i
i 1
Fall 2010
wˆ  w as N   with probability 1, where
For stable systems:
w is called the long-run average system time..
If the system under consideration is the queue alone (with WiQ time
N
1
customer i spends in the queue): wˆ Q  WiQ  wQ as N  
N i 1
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4. Long-Run Measures (3):
Average Time Spent in System Per Customer w (2)
G/G/1/N/K example (cont.):
The average system time is:
W1  W2  ...  W5 2  (8  3)  ...  (20  16)

 4.6 time units
5
5
Assumptions are single server, FIFO queue discipline.
ˆ
w
Time spent in waiting line is:
W  W2  ... W5
0  0 33 0
ˆwQ  1

 1.2 timeunits
5
5
Fall 2010
Q
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Q
Q
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4. Long-Run Measures (4): Little’s Law (1)
Conservation equation (a.k.a. Little’s law)
Average # in
system
Lˆ  lˆwˆ
Average
System time
Arrival rate
L  lw as T   and N  
• Holds for almost all queueing systems or subsystems
(regardless of the number of servers, the queue discipline, or
other special circumstances).
Fall 2010
• G/G/1/N/K example (cont.): On average, one arrival every 4
time units and each arrival spends 4.6 time units in the system.
Hence, at an arbitrary point in time, there is (1/4)(4.6) = 1.15
customers present on average.
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4. Long-Run Measures (4): Little’s Law (2)
Total system time for all customers is given by the total area under
the number in system function L(t).
N
W  
i 1
T
i
0
L(t )dt
Therefore,
Fall 2010
N
T
1
1
N
ˆwˆ
Lˆ   L(t )dt 
W

l

i
0
T
T N i 1
where lˆ  N T
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Little’s Law
25
4. Long-Run Measures (5):
Server Utilization (1)
Definition: the proportion of time that a server is busy.
• Observed server utilization, rˆ, is defined over a specified time
interval [0,T].
• Long-run server utilization is r.
Fall 2010
• For systems with long-run stability:
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rˆ  r as T  
26
4. Long-Run Measures (5): Server
Utilization (2)
As per the above picture, the server utilization is
rˆ  (totalbusy time)/ T 
 T  T  T  T  T  17 / 20

i 1 i
0
Fall 2010
T0 is the totalidle time
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4. Long-Run Measures (5): Server
Utilization (3)
For G/G/1/∞/∞ queues:
• Any single-server queueing system with average arrival
rate l customers per time unit, where average service time
E(S) = 1/m time units or m is the average service rate ,
infinite queue capacity and calling population.
• Conservation equation, L = lw, can be applied.
• For a stable system, the average arrival rate to the server,
ls, must be identical to l.
• The average number of customers in the server is:
Fall 2010
ˆL  1
s
T
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
T
0

T  T0
L(t )  LQ (t ) dt 
T
For a single-server case,
the average number of customers being served at any
arbitrary point in time is equal to server utilization!!
rˆ
What is this??
28
4. Long-Run Measures (5): Server
Utilization (4)
Fall 2010
The actual number of customers in the server subsystem is either 0 or 1
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4. Long-Run Measures (5): Server
Utilization (5)
• In general, for a single-server queue:
Lˆs  rˆ  Ls  r as T  
and combiningthisinto L  lw, for theserversub system, r  lE ( s) 
l
m
– For a single-server stable queue:
· The arrival rate l must be less than the service rate m; l < m
r
– For an unstable queue (l > m)
l
<1
m
Fall 2010
· The server is always busy
· Waiting line tend to grow in length at an average rate of l m
customers per time unit and long run average queue length is 
· long-run server utilization is 1.
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4. Long-Run Measures (5): Server
Utilization (6)
For G/G/c/∞/∞ queues:
• A system with c identical servers in parallel. Arrivals occur
at a rate l and each server works at a rate m customers per
time unit.
• If an arriving customer finds more than one server idle, the
customer chooses a server without favoring any particular
server.
• For systems in statistical equilibrium, the average number of
busy servers, Ls, is: Ls, = lE(s) = l/m.
• The long-run average server utilization is:
Fall 2010
Ls
l
r

, where l < cm for stablesystems
c cm
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4. Long-Run Measures (5): Server
Utilization (7)
Customers arrive at random to a license bureau at a rate of l=50
customers per hour. Currently, there are 20 clerks, each
serving m=5 customers per hour on the average.
Therefore, the long-run or steady state average utilization of a
server is
r
l
50

 0.5
cm 20(5)
The average number of busy servers is:
Ls 
A typical clerk is busy only 50% of the time!!
l 50
  10
m 5
Can we decrease the number of servers??
• For system to be stable, the number of servers must be
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c>
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l 50

 10
m 5
Possibilities are c11, 12 etc
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4. Long-Run Measures (5): Server
Utilization vs. System Performance (8)
System performance varies widely for a given utilization r.
• For example, a D/D/1 queue where E(A) = 1/l and E(S) = 1/m,
where:
L = r = l/m, w = E(S) = 1/m, LQ = WQ = 0.
– By varying l and m, server utilization can assume any value
between 0 and 1.
– Yet there is never any line.
Fall 2010
• In general, variability of inter-arrival and service times causes
lines to fluctuate in length.
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4. Long-Run Measures (5): Server
Utilization vs. System Performance (9)
Example: A physician who schedules patients every 10 minutes and spends Si minutes
with the ith patient:
 9 minuteswith probability 0.9
Si  
12 minuteswith probability 0.1
• Arrivals are deterministic, A1 = A2 = … = l-1 = 10.
• Services are stochastic, E(Si) = 9.3 min and V(Si) = 0.81 min2.
• On average, the physician's utilization = r  l/m = 0.93 < 1.
• Consider the system is simulated with service times: S1 = 9, S2 = 12, S3 =
9, S4 = 9, S5 = 9, …. The system becomes:
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• The occurrence of a relatively long service time (S2 = 12) causes a waiting
line to form temporarily.
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4. Long-Run Measures (6): Costs in
Queueing Problems
Costs can be associated with various aspects of the waiting line or
servers:
• System incurs a cost for each customer in the queue, say at a
rate of $10 per hour per customer.
– The average cost per customer is:
N

j 1
$10*W jQ
N
 $10* wˆ Q
WjQ is the time
customer j spends
in queue
– If lˆ customers per hour arrive (on average), the
average cost per hour is:
ˆQ 
 ˆ customer $10* w
  $10* lˆw
ˆ Q  $10* LˆQ / hour
l


hour  customer

• Server may also impose costs on the system, if a group of c
parallel servers (1  c  ∞) have utilization r, each server
imposes a cost of $5 per hour while busy.
Fall 2010
– The total server cost is: $5*cr.
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5. Steady-State Behavior of InfinitePopulation Markovian Models (1)
Markovian models: exponential-distribution arrival process
(mean arrival rate = l).
Service times may be exponentially distributed (M) or
arbitrary (G).
A queueing system is in statistical equilibrium if the
probability that the system is in a given state is not time
dependent:
P( L(t) = n ) = Pn(t) = Pn.
Fall 2010
Mathematical models in this chapter can be used to obtain
approximate results even when the model assumptions do
not strictly hold (as a rough guide).
Simulation can be used for more refined analysis (more faithful
representation for complex systems).
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5. Steady-State Behavior of InfinitePopulation Markovian Models (2)
For the simple model studied in this chapter, the steady-state
parameter, L, the time-average number of customers in the system

is:
L
 nP
n
n 0
• Apply Little’s equation to the whole system and to the queue
alone:
L
1
w  , wQ  w 
l
m
LQ  lwQ
Fall 2010
G/G/c/∞/∞ example: to have a statistical equilibrium, a necessary
and sufficient condition is l/(cm) < 1.
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5. Steady-State Behavior of Infinite-Population
Markovian Models (3): M/G/1 Queues (1)
Single-server queues with Poisson arrivals & unlimited capacity.
Suppose service times have mean 1/m and variance s2 and r = l/m < 1, the
steady-state parameters of M/G/1 queue:
r  l / m , P0  (1  r ) is theprobability serveris idle (no customers)
l2 (1 m 2  s 2 )
r 2 (1  s 2 m 2 )
Lr
r
2(1  r )
2(1  r )
r 2 (1  s 2 m 2 )
LQ 
2(1  r )
l (1 / m 2  s 2 )
w 
m
2(1  r )
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1
l (1 / m 2  s 2 )
wQ 
2(1  r )
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5. Steady-State Behavior of Infinite-Population
Markovian Models (3): M/G/1 Queues (2)
• No simple expression for the steady-state probabilities P0, P1, …
• L – LQ = r is the time-average number of customers being served.
• Average length of queue, LQ, can be rewritten as:
r2
l2s 2
LQ 

2(1  r ) 2(1  r )
Fall 2010
– If l and m are held constant, LQ depends on the variability, s2,
of the service times.
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5. Steady-State Behavior of Infinite-Population
Markovian Models (3): M/G/1 Queues (3)
Example: Widget making machines malfunction at random and then
require mechanic’s attention. Malfunctions occur according to a poisson
process at a rate l = 1.5 per hour. Observation over several months has
found that repair times by the single mechanic take an average time of 30
minutes with a standard deviation of 20 minutes.
Therefore, the mean service time is 1/m = ½ =0.5 hour, the service rate m =
2 per hour and the variance s2 = 202 minutes = 1/9 hour.
The customers are the widget makers and appropriate model is M/G/1
because only the mean and variance of the service times are known, not the
distribution
Therefore, r=l/m=1.5/2 = 0.75.
Steady state time average number of broken machines is:
Fall 2010
(1.5) 2 [0.52  1 / 9]
L  0.75 
 2.375customers
2(1  0.75)
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5. Steady-State Behavior of Infinite-Population
Markovian Models (3): M/G/1 Queues (4)
Example: Two workers competing for a job, Able claims to be faster than Baker on
average, but Baker claims to be more consistent,
• Poisson arrivals at rate l = 2 per hour (1/30 per minute).
• Able: 1/m = 24 minutes and s2 = 202 = 400 minutes2:
(1 / 30) 2 [242  400]
LQ 
 2.711customers
2(1  4 / 5)
– The proportion of arrivals who find Able idle and thus experience no delay is P0
= 1-r = 1/5 = 20%.
• Baker: 1/m = 25 minutes and s2 = 22 = 4 minutes2:
Fall 2010
(1 / 30) 2 [252  4]
LQ 
 2.097 customers
2(1  5 / 6)
– The proportion of arrivals who find Baker idle and thus experience no delay is
P0 = 1-r = 1/6 = 16.7%.
• Although working faster on average, Able’s greater service variability results in an
average queue length about 30% greater than Baker’s.
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5. Steady-State Behavior of Infinite-Population
Markovian Models (4): M/M/1 Queues (1)
Suppose the service times in an M/G/1 queue are exponentially
distributed with mean 1/m, then the variance is s2 = 1/m2.
• M/M/1 queue is a useful approximate model when service
times have standard deviation approximately equal to their
means.
• The steady-state parameters:
r  l / m , P0  (1- r ), Pn  1  r r n
l2
r2
L

, LQ 

m  l 1 r
m m  l  1  r
1
1
l
r
w

, wQ 

m  l m (1  r )
m m  l  m (1  r )
Fall 2010
l
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r
42
5. Steady-State Behavior of Infinite-Population
Markovian Models (4): M/M/1 Queues (2)
State transition diagram of M/M/1 queue
l
0
l
1
m
l
k-1
2
m
l
l
k
m
m
l
m
l
k+1
m
m
Flow Rat e int o Ek  lk 1 Pk 1  m k 1 Pk 1
Flow Rat e out of Ek  lk Pk  m k Pk
In equilibrium, t hese t wo must be t hesame  lk 1 Pk 1  m k 1 Pk 1  lk  m k Pk
For M/M/1: lk  l ; m k  m
Fall 2010
l
P;
m 0
k
l
l
l
l
P1  2 P0 ;; Pk  k P0    P0
m
m
m
m
l

W e also know t hatk 0 Pk  1
 P0  1   1  r
m
 P1 
 Pk  1  r r k
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P2 
2
k
43
5. Steady-State Behavior of Infinite-Population
Markovian Models (4): M/M/1 Queues (3)
Solution for Pk
1-r
(1-r)r2
Pk
(1-r)r3
k
The average number in the system M/M/1
Fall 2010
L
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r
1
44
M/M/1 Queues
[Steady-State of
Markovian Model]
Example: M/M/1 queue with service rate m10 customers per hour.
• Consider how L and w increase as arrival rate, l, increases from 5
to 8.64 by increments of 20%:
l
r
L
w
5.0
6.0
7.2
8.64
10.0
0.500
0.600
0.720
0.864
1.000
Fall 2010
∞
0.20
0.25
0.36
0.73
∞
• If l/m  1, waiting lines tend to continually grow in length.
• Increase in average system time (w) and average number in system
(L) is highly non-linear as a function of r.
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1.00
1.50
2.57
6.35
45
5. Steady-State Behavior of Infinite-Population
Markovian Models (4): M/M/1 Queues (4)
Example: If arrivals are occurring at rate l = 10 per hour.
Management has a choice of two servers, one who works at rate m1=11
customers per hour and second at rate m2=12 customers per hour
Respective utilizations are:
• r1=l/m1=10/11=0.909; r2=l/m2=10/12=0.833
Average number in the system is:
• L1= r1/(1- r1)=10; L2= r2/(1- r2)=5
Fall 2010
Thus, a decrease in service rate from 12 to 11 customers per hour, a
mere 8.3% decrease would result in an increase in average number in
system from 5 to 10, which is 100% increase!!!
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5. Steady-State Behavior of Infinite-Population
Markovian Models (5): Effect of Utilization and
Service Variability (1)
For almost all queues, if lines are too long, they can be reduced by
decreasing server utilization (r) or by decreasing the service time
variability (s2).
A measure of the variability of a distribution is coefficient of variation
(cv):
V (X )
SD
(cv) 
or cv 
2
E( X )
E ( X )
2
• The larger cv is, the more variable is the distribution relative to its
expected value
For deterministic servers V(X)=0  cv = 0
Fall 2010
For exponential servers, E(X)=1/m; V(X)=1/m2  cv = 1;
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47
5. Steady-State Behavior of Infinite-Population
Markovian Models (5): Effect of Utilization and
Service Variability (2)
Consider LQ for any M/G/1
queue:
r 2 (1  s 2 m 2 )
LQ 
2(1  r )
 r 2  1  (cv) 2 



 1  r 
2 


Fall 2010
LQ for M/M/1
queue
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Corrects the M/M/1
formula to account
for a non-exponential
service time dist’n
48
5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (1)
M/M/c/∞/∞ queue: c channels operating in parallel.
• Each channel has an independent and identical exponential
service-time distribution, with mean 1/m.
• To achieve statistical equilibrium, the offered load (l/m) must
satisfy l/m < c, where l/(cm) = r is the server utilization.
• Some of the steady-state probabilities are given in terms of
– P0, the probability that the system is empty
– P(L()c) where L() is a random variable representing the
number in the system in statistical equilibrium
Fall 2010
· Implies the probability of all servers busy or
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

n c
Pn
49
5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (2)
M/M/c/∞/∞ queue: c channels operating in parallel:
r  l / cm
 c 1 (l / m ) n   l  c  1  cm  
 
P0  
     
n
!
m
c
!
c
m

l



 
  
 n 0
 c 1 (cr ) n  

c  1  1   




  

c
r


 

 
 n 0 n!  
 c!  1  r  
1
1
c


l m c P0
cr  P0
P L   c  

c!1  l cm  c!1  r 
(cr ) c 1 P0
r P  L ( )  c 
L  cr 
 cr 
c(c!)(1  r ) 2
w
1 r
L
l
wQ  w 
1
m
Fall 2010
(cr ) c 1 P0
r P  L ( )  c 
LQ  lwQ 

2
c(c!)(1  r )
1 r
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L  LQ 
l
 cr
m
50
5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (3)
Fall 2010
P0 Steady state probability that system is empty depends only on c & r
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51
5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (4)
Example: Attendants manage the tool cribs, as mechanics, assumed to be
from an infinite calling population, arrive for service
Assume Poisson arrivals at rate 2 mechanics per minute and exponentially
distributed service times with mean 40 seconds
l=2 per minute m=60/40=1.5 per minute  offered load > 1
Let there be c=2 attendants
Probability P0 = 0.2
Probability that all servers are busy P(L()2) = 0.533
Time average length of waiting line LQ = 1.07 mechanics
Time average number in the system L = LQ+l/m = 2.4 mechanics
From little’s formula w = L/l = 1.2 minutes
Average time spent waiting wQ = w – 1/m = 0.533 minutes
Fall 2010
Server utilization l/(cm) = r = 0.667
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5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (5)
Other common multiserver queueing models:
• M/G/c/∞: general service times and c parallel server. The
parameters can be approximated from those of the M/M/c/∞/∞
model.
– Use the similar correction factor (1+cv2)/2 that was used in M/G/1
• M/G/∞: general service times and infinite number of servers,
e.g., customer is its own system, service capacity far exceeds
service demand
Fall 2010
• M/M/C/N/∞: service times are exponentially distributed at rate
m and c servers where the total system capacity is N  c customer
(when an arrival occurs and the system is full, that arrival is
turned away).
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53
5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (6)
Other common multiserver queueing models:
• M/G/c/∞: general service times and c parallel server. The
parameters can be approximated from those of the M/M/c/∞/∞
model.
– Use the similar correction factor (1+cv2)/2 that was used in M/G/1
• M/G/∞: general service times and infinite number of servers,
e.g., customer is its own system, service capacity far exceeds
service demand
P0  e  l / m
w  1 ; wQ  0
m
Ll
Fall 2010
Pn 
CSC 446/546
e
m ; LQ  0
l / m
n
 l 
 m  , n  0,1,...
n!
54
5. Steady-State Behavior of Infinite-Population
Markovian Models (6): Multiserver Queue (7)
M/M/c/N/∞ queue: c channels operating in parallel; Limited Capacity (N 
c), when an arrival occurs and the system is full, that arrival is turned away.
Service times are exponentially distributed.
r  l / cm ;
a l/m
1
c

a n a c N n c 
P0  1   
r 
 n 0 n! c! n c 1

a N P0
PN 
c!c N c
ra c P0
LQ 
1  r N c  ( N  c) r N c (1  r )
2
c!1  r 
le  l (1  PN )

wQ 

LQ
le
w  wQ 
1
m
Fall 2010
L  le .w
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55
6. Steady-State Behavior of FinitePopulation Models (1)
When the calling population is small,
the presence of one or more
customers in the system has a
strong effect on the distribution of
future arrivals.
Fall 2010
Consider a finite-calling population
model with K customers
(M/M/c/K/K):
• The time between the end of one
service visit and the next call for
service is exponentially
distributed, (mean = 1/l).
• Service times are also
exponentially distributed.
• c parallel servers and system
capacity is K.
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56
6. Steady-State Behavior of FinitePopulation Models (2)
• Some of the steady-state probabilities for (M/M/c/K/K):
n
 c 1  K  l  n K
 l  
K!
 
P0      
n c   
n
m
 n 0    n c ( K  n)!c!c  m  
 K  l  n
   P0 ,
n  0,1,..., c  1
 n m
Pn    
n


K
!
l

 ( K  n)!c!c n c  m  , n  c, c  1,... K
 



K
L
1
 nP ,
w  L / le ,
n
r  le / cm
n 0
wherele is the longrun effective arrivalrate of customersto queue(or entering/exitingservice)
le 
K
 ( K  n )lP
n
Fall 2010
n 0
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57
6. Steady-State Behavior of FinitePopulation Models (3)
Example: two workers are responsible for10 milling machines.
• Machines run on the average for 20 minutes, then require an
average 5-minute service period, both times exponentially
distributed: l = 1/20 and m = 1/5.
• All of the performance measures depend on P0:
1
n
 21 10 5  n 10
10!
 5  
P0      
    0.065
n2
n
20
 n 0    n  2 (10  n)!2!2  20  


– Then, we can obtain the other Pn.
– Expected number of machines in system:
10
L
 nP  3.17 machines
n
n 0
– The average number
of running machines:
Fall 2010
K  L  10  3.17  6.83 machines
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58
7. Final Summary
Introduced basic concepts of queueing models.
Show how simulation, and some times mathematical analysis, can
be used to estimate the performance measures of a system.
Commonly used performance measures: L, LQ, w, wQ, r, and le.
When simulating any system that evolves over time, analyst must
decide whether to study transient behavior or steady-state
behavior.
• Simple formulas exist for the steady-state behavior of some
queues.
Fall 2010
Simple models can be solved mathematically, and can be useful in
providing a rough estimate of a performance measure.
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59