[PPT] - Simulation Simulation Modeling and Performance Analysis with PowerPoint Presentation

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Computer Science, Informatik 4 Communication and Distributed Systems

Simulation Simulation

Modeling and Performance Analysis with Discrete-Event Simulation g y

Dr. Mesut Güneş

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Computer Science, Informatik 4 Communication and Distributed Systems

Chapter 8

Queueing Models

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Computer Science, Informatik 4 Communication and Distributed Systems

Contents Contents Characteristics of Queueing Systems Characteristics of Queueing Systems Queueing Notation – Kendall Notation Long-run Measures of Performance of Queueing Systems g g y Steady-state Behavior of Infinite-Population Markovian Models Steady-state Behavior of Finite-Population Models Networks of Queues

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Chapter 8. Queueing Models 3

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Purpose Purpose

Simulation is often used in the analysis of queueing models.

Simulation is often used in the analysis of queueing models.

A simple but typical queueing model

S W i i li Calling population

Queueing models provide the analyst with a powerful tool for

Server Waiting line

Queueing models provide the analyst with a powerful tool for designing and evaluating the performance of queueing systems.

Typical measures of system performance

S tili ti l th f iti li d d l f t

Server utilization, length of waiting lines, and delays of customers
For relatively simple systems, compute mathematically
For realistic models of complex systems, simulation is usually required
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Chapter 8. Queueing Models 4

p y y q

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Outline Outline Discuss some well-known models Discuss some well known models

Not development of queueing theory, for this see other class!

We will deal with

General characteristics of queues
Meanings and relationships of important performance measures
Estimation of mean measures of performance
Effect of varying input parameters
Mathematical solutions of some basic queueing models
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Characteristics of Queueing Systems Characteristics of Queueing Systems

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Characteristics of Queueing Systems Characteristics of Queueing Systems Key elements of queueing systems Key elements of queueing systems

Customer: refers to anything that arrives at a facility and requires

service, e.g., people, machines, trucks, emails. S f t th t id th t d i

Server: refers to any resource that provides the requested service, e.g.,

repairpersons, retrieval machines, runways at airport. System Customers Server System Customers Server Reception desk People Receptionist Hospital Patients Nurses Airport Airplanes Runway Production line Cases Case-packer R d k C T ffi li h Road network Cars Traffic light Grocery Shoppers Checkout station Computer Jobs CPU disk CD

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Chapter 8. Queueing Models 7

Computer Jobs CPU, disk, CD Network Packets Router

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Calling Population Calling Population

Calling population: the population of potential customers, may be

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 customers being served and waiting, e.g., model of one corporate jet, if it is being repaired, the repair arrival rate becomes zero.

n n-1

Infinite population model: if arrival rate is not affected by the number of

customers being served and waiting e g systems with large population customers being served and waiting, e.g., systems with large population

f potential customers.

∞

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Chapter 8. Queueing Models 8

∞

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System Capacity System Capacity System Capacity: a limit on the number of customers that may 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

t it i li t t th h i to wait in line to enter the mechanism.

Unlimited capacity e g

concert ticket sales with no limit on the number

Server Waiting line

Unlimited capacity, e.g., concert ticket sales with no limit on the number
f people allowed to wait to purchase tickets.
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Chapter 8. Queueing Models 9

Server Waiting line

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Arrival Process Arrival Process

For infinite-population models:

For infinite population models:

In terms of interarrival times of successive customers.
Random arrivals: interarrival times usually characterized by a probability

distribution distribution.

Most important model: Poisson arrival process (with rate λ), where An

represents the interarrival time between customer n-1 and customer n, and is exponentially distributed (with mean 1/λ) exponentially distributed (with mean 1/λ).

Scheduled arrivals: interarrival times can be constant or constant plus or

minus a small random amount to represent early or late arrivals minus a small random amount to represent early or late arrivals.

Example: 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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Arrival Process Arrival Process

For finite-population models:

For finite population models:

Customer is pending when the customer is outside the queueing system,

e.g., machine-repair problem: a machine is “pending” when it is

perating it becomes “not pending” the instant it demands service from
perating, it becomes not pending the instant it demands service from

the repairman.

Runtime of a customer is the length of time from departure from the

queueing system until that customer’s next arrival to the queue e g 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 (TTF).

Let A (i) A (i)

be the successive runtimes of customer i and S (i) S (i)

Let A1

(i), A2 (i), … be the successive runtimes of customer i, and S1 (i), S2 (i)

be the corresponding successive system times:

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Queue Behavior and Queue Discipline Queue Behavior and Queue Discipline Queue behavior: the actions of customers while in a queue Queue behavior: 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

R l ft b i i th li h it i t l l

Renege: leave after being in the line when its moving too slowly
Jockey: move from one line to a shorter line

Queue discipline: 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)

Last-in-first-out (LIFO)

Service in random order (SIRO)
Shortest processing time first (SPT)
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Service according to priority (PR)

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Service Times and Service Mechanism Service Times and Service Mechanism

Service times of successive arrivals are denoted by S1, S2, S3.

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 identically distributed random variables, e.g., exponential, Weibull, gamma, lognormal, and truncated normal distribution.

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

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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Service Times and Service Mechanism Service Times and Service Mechanism

Example: consider a discount warehouse where customers may:

Example: consider a discount warehouse where customers may:

Serve themselves before paying at the cashier
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Service Times and Service Mechanism Service Times and Service Mechanism

Wait for one of the three clerks:
Batch service (a server serving several customers simultaneously), or

customer requires several servers simultaneously.

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Service Times and Service Mechanism Service Times and Service Mechanism

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Example Example Candy production line Candy production line

Three machines separated by buffers
Buffers have capacity of 1000 candies

Assumption:Allways sufficient supply of

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Chapter 8. Queueing Models 17

pp y raw material.

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Queueing Notation – Kendall Notation Queueing Notation Kendall Notation

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Queueing Notation – Kendall Notation Queueing Notation – Kendall Notation

A notation system for parallel server queues: A/B/c/N/K

y p q

A

represents the interarrival-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

N, K are usually dropped, if they are infinity
Common symbols for A and B
M

Markov, exponential distribution

D

Constant, deterministic E E l di t ib ti f d k

Ek

Erlang distribution of order k

H

Hyperexponential distribution

G

General, arbitrary

Examples
Examples
M/M/1/∞/∞ same as M/M/1: Single-server with unlimited capacity and call-
population. Interarrival and service times are exponentially distributed
G/G/1/5/5: Single-server with capacity 5 and call-population 5.
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G/G/1/5/5: Single server with capacity 5 and call population 5.

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Queueing Notation Queueing Notation

General 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

λ

arrival rate

λe

effective arrival rate

μ

service rate of one server

ρ

server utilization A i t i l ti b t t 1 d

An

interarrival time between customers n-1 and n

Sn

service time of the n-th arriving customer

Wn

total time spent in system by the n-th arriving customer W Q total time spent in the waiting line by customer

Wn

Q

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

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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wQ long run average time spent in queue per customer

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Long-run Measures of Performance of Queueing Systems Long run Measures of Performance of Queueing Systems

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Long-run Measures of Performance of Queueing Systems Long-run Measures of Performance of Queueing Systems

Primary long-run measures of performance are Primary long run measures of performance are

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

w

long-run average time spent in system per customer

wQ

long-run average time spent in queue per customer

ρ

server utilization

Other measures of interest are

Long-run proportion of customers who are delayed longer than t0

time units time units

Long-run proportion of customers turned away because of

capacity constraints

Long-run proportion of time the waiting line contains more than

k0 customers

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Long-run Measures of Performance of Queueing Systems Long-run Measures of Performance of Queueing Systems

Goal of this section Goal of this section

Major measures of performance for a general G/G/c/N/K

queueing system

How these measures can be estimated from simulation runs

Two types of estimators Two types of estimators

Sample average
Time-integrated sample average

g p g

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Time-Average Number in System L Time-Average Number in System L

Consider a queueing system over a period of time T

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: defined by:

∑ ∑

∞ = ∞ =

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = 1 ˆ

i i i i

T T i iT T L

Consider the total area under the function is L(t), then,

∫ ∑

∞ T

1 1 ˆ

∫ ∑

= =

= T i i

dt t L T iT T L ) ( 1 1 ˆ

The long-run time-average number of customers in system, with

probability 1:

) ( 1 ˆ L dt t L L

T

⎯ ⎯ → ⎯ = ∫

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) ( L dt t L T L

T ⎯

⎯ → ⎯ =

∞ →

∫

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Time-Average Number in System L Time-Average Number in System L

Number of customers in the system y Time

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Time-Average Number in System L Time-Average Number in System L The time-weighted-average number in queue is: The time weighted average number in queue is:

Q T T Q i Q i Q

L dt t L T iT T L ⎯ ⎯ → ⎯ = =

∞ → ∞

∫ ∑

) ( 1 1 ˆ

G/G/1/N/K example: consider the results from the queueing system (N ≥ 4,

K ≥ 3).

i

T T

=0

customers 15 1 20 / 23 20 / )] 1 ( 3 ) 4 ( 2 ) 12 ( 1 ) 3 ( [ ˆ = = + + + = L

⎩ ⎨ ⎧ ≥ − = = 1 if 1 ) ( if , ) ( L(t) t L L(t) t LQ

customers 15 . 1 20 / 23 = =

⎩ ≥1 if , 1 ) ( L(t) t L

customers 3 . 20 ) 1 ( 2 ) 4 ( 1 ) 15 ( ˆ = + + =

Q

L

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20

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Average Time Spent in System Per Customer w

The average time spent in system per customer, called the average

Average Time Spent in System Per Customer w

The average time spent in system per customer, called the average system time, is:

∑

=

N i

W w 1 ˆ

where W1, W2, …, WN are the individual times that each of the N

∑

= i i

W N w

1

customers spend in the system during [0,T].

For stable systems:

∞ → → N w w as ˆ

If the system under consideration is the queue alone:

N

1 ∑

= ∞ →

⎯ ⎯ → ⎯ =

N i Q N Q i Q

w W N w

1

1 ˆ

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Average Time Spent in System Per Customer w Average Time Spent in System Per Customer w

G/G/1/N/K example (cont.):

p ( )

The average system time is

i i 6 4 ) 16 20 ( ... ) 3 8 ( 2 ... ˆ

5 2 1

− + + − + + + + W W W

The average queuing time is

units time 6 . 4 5 ) 16 20 ( ... ) 3 8 ( 2 5 ... ˆ

5 2 1

= + + + = + + + = W W W w

3 3 + + + + units time 2 . 1 5 3 3 ˆ = + + + + =

Q

w

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The Conservation Equation – Little’s Law The Conservation Equation – Little s Law

Conservation equation (a.k.a. Little’s law)

Conservation equation (a.k.a. Little s law)

w L ˆ ˆ ˆ λ =

Average System time Average # in t Arrival rate System time system

Holds for almost all queueing systems or subsystems (regardless of the

∞ → ∞ → = N T w L and as λ

Holds for almost all queueing systems or subsystems (regardless of the

number of servers, the queue discipline, or other special circumstances).

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 and each arrival spends 4.6 time units in the system. Hence, at an arbitrary point in time, there are (1/4)(4.6) = 1.15 customers present on average.

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Server Utilization Server Utilization

Definition: the proportion of time that a server is busy.

Definition: the proportion of time that a server is busy.

Observed server utilization, , is defined over a specified time interval

[0,T ].

Long run server utilization is ρ

ρ ˆ

Long-run server utilization is ρ.
For systems with long-run stability:

∞ → → T as ˆ ρ ρ

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Server Utilization Server Utilization

For G/G/1/∞/∞ queues:

For G/G/1/∞/∞ queues:

Any single-server queueing system with average arrival rate λ

customers per time unit, where average service time E(S) = 1/μ ti it i fi it it d lli l ti time units, infinite queue capacity and calling population.

Conservation equation, L = λw, can be applied.
For a stable system, the average arrival rate to the server, λ ,

For a stable system, the average arrival rate to the server, λs, must be identical to λ.

The average number of customers in the server is:

( )

T T T dt t L t L T L

T Q s

) ( ) ( 1 ˆ − = − = ∫ T T

∫

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Server Utilization Server Utilization

In general, for a single-server queue:

In general, for a single server queue:

ρ ρ = ⎯ ⎯ → ⎯ =

∞ →

ˆ ˆ L L

s T s

μ λ λ ρ = ⋅ = ) ( and s E

For a single-server stable queue:

1 < = μ λ ρ

For an unstable queue (λ > μ), long-run server utilization is 1.

μ

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Server Utilization Server Utilization

For G/G/c/∞/∞ queues:

For G/G/c/∞/∞ queues:

A system with c identical servers in parallel.
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, L , is: servers, Ls, is:

Clearly 0 ≤ LS ≤ c

μ λ λ = = ) (S E LS

The long-run average server utilization is:

λ L systems stable for where , μ λ μ λ ρ c c c Ls < = =

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Server Utilization and System Performance Server Utilization and System Performance

System performance varies widely for a given utilization ρ.

y p y g ρ

For example, a D/D/1 queue where E(A) = 1/λ and E(S) = 1/μ,

where:

L λ/ E(S) 1/ L W L = ρ = λ/μ, w = E(S) = 1/μ, LQ = WQ = 0

By varying λ and μ, server utilization can assume any value

between 0 and 1.

Yet there is never any line.
In general, variability of interarrival and service times causes

lines to fluctuate in length. lines to fluctuate in length.

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Server Utilization and System Performance Server Utilization and System Performance

Example: A physician who
Consider the system is simulated

p p y schedules patients every 10 minutes and spends Si minutes with the i-th patient: y with service times: S1= 9, S2=12, S3 = 9, S4 = 9, S5 = 9, ….

The system becomes:

y

⎩ ⎨ ⎧ = 1 . y probabilit with minutes 12 9 . y probabilit with minutes 9

i

S

Arrivals are deterministic,

A1 = A2 = … = λ-1 = 10. S i t h ti

Services are stochastic
E(Si) = 9.3 min
V(S0) = 0.81 min2
σ = 0.9 min
On average, the physician's

utilization = ρ = λ/μ = 0.93 < 1.

The occurrence of a relatively

long service time (S2 = 12) causes a waiting line to form

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Chapter 8. Queueing Models 35

causes a waiting line to form temporarily.

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Costs in Queueing Problems Costs in Queueing Problems

Costs can be associated with various aspects of the waiting line or

p g servers:

System incurs a cost for each customer in the queue, say at a rate of $10

per hour per customer.

W Q is the time

p p

The average cost per customer is:

Q N Q j

w W ˆ 10 $ 10 $ ⋅ = ⋅

∑

Wj

Q is the time

customer j spends in queue

If customers per hour arrive (on average), the average cost per

hour is:

Q j

w N 10 $

1

∑

=

λ ˆ

hour is:

hour ˆ 10 $ ˆ ˆ 10 $ customer ˆ 10 $ hour customer ˆ

Q Q Q

L w w ⋅ = ⋅ ⋅ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ λ

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.

⎠ ⎝

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Chapter 8. Queueing Models 36

y

The total server cost is:

ρ ⋅ ⋅c 5 $

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Steady-state Behavior of Infinite-Population Markovian Models Steady state Behavior of Infinite Population Markovian Models

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Steady-State Behavior of Markovian Models Steady-State Behavior of Markovian Models

Markovian models: Markovian models:

Exponential-distributed arrival process (mean arrival rate = 1/λ).
Service times may be exponentially (M) or arbitrary (G) distributed.

Q di i li i FIFO

Queue discipline is FIFO.
A queueing system is in statistical equilibrium if the probability that the

system is in a given state is not time dependent:

n n

P t P n t L P = = = ) ( ) ) ( (

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
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Chapter 8. Queueing Models 38

representation for complex systems.

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Steady-State Behavior of Markovian Models Steady-State Behavior of Markovian Models

Properties of processes with statistical equilibrium Properties of processes with statistical equilibrium

The state of statistical equilibrium is reached from any starting

state.

The process remain in statistical equilibrium once it has reached

it.

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Steady-State Behavior of Markovian Models Steady-State Behavior of Markovian Models

For the simple model studied in this chapter, the steady-state For the simple model studied in this chapter, the steady state parameter, L, the time-average number of customers in the system is:

∞

Apply Little’s equation L=λ w to the whole system and to the

∑

=

n n

nP L

Apply Little s equation, L=λ w, to the whole system and to the

queue alone:

Q Q Q

w L w w L w λ λ = − = = , 1 ,

G/G/c/∞/∞ example: to have a statistical equilibrium, a d ffi i t diti i

Q Q Q

μ λ

necessary and sufficient condition is:

1 < = λ ρ

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μ ρ c

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M/G/1 Queues M/G/1 Queues

Single-server queues with Poisson arrivals and unlimited capacity.

S g e se e queues

sso

a a s a d u ed capac y

Suppose service times have mean 1/μ and variance σ 2 and ρ = λ / μ < 1, the

steady-state parameters of M/G/1 queue:

λ 1 ρ μ λ ρ − = = P ) 1 ( 2 ) 1 ( 1

2 2 2

ρ μ σ ρ ρ ρ − + + = L P

The particular distribution is not known!

) 1 ( 2 ) / 1 ( 1

2 2

ρ σ μ λ μ − + + = w ) / 1 ( ) 1 ( 2 ) 1 (

2 2 2 2 2

λ ρ μ σ ρ − + =

Q

L

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) 1 ( 2 ) / 1 (

2 2

ρ σ μ λ − + =

Q

w

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M/G/1 Queues M/G/1 Queues

There are no simple expressions for the steady-state probabilities P0,

There are no simple expressions for the steady state probabilities P0, P1, P2 ,…

L – LQ = ρ is the time-average number of customers being served.
Average length of queue, LQ, can be rewritten as:

) 1 ( 2 ) 1 ( 2

2 2 2

ρ σ λ ρ ρ − + − =

Q

L

If λ and μ are held constant, LQ depends on the variability, σ 2, of the

service times.

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M/G/1 Queues M/G/1 Queues

Example: Two workers competing for a job, Able claims to be faster than

B k b B k l i b i Baker on average, but Baker claims to be more consistent,

Poisson arrivals at rate λ = 2 per hour (1/30 per minute).
Able: 1/μ = 24 minutes and σ 2 = 202 = 400 minutes2:

customers 711 . 2 ) 5 / 4 1 ( 2 ] 400 24 [ ) 30 / 1 (

2 2

= − + =

Q

L

The proportion of arrivals who find Able idle and thus experience no delay is

P0 = 1-ρ = 1/5 = 20%.

Baker: 1/μ = 25 minutes and σ 2 = 22 = 4 minutes2:

Baker: 1/μ 25 minutes and σ 2 4 minutes :

customers 097 . 2 ) 6 / 5 1 ( 2 ] 4 25 [ ) 30 / 1 (

2 2

= − + =

Q

L

The proportion of arrivals who find Baker idle and thus experience no delay is

P0 = 1-ρ = 1/6 = 16.7%.

Although working faster on average Able’s greater service variability results in an
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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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Computer Science, Informatik 4 Communication and Distributed Systems

M/M/1 Queues M/M/1 Queues

Suppose the service times in an M/G/1 queue are exponentially

Suppose the service times in an M/G/1 queue are exponentially distributed with mean 1/μ, then the variance is σ 2 = 1/μ2.

M/M/1 queue is a useful approximate model when service times

h t d d d i ti i t l l t th i have standard deviation approximately equal to their means.

The steady-state parameters

λ

( )

1 ρ λ ρ ρ μ ρ − = =

n n

P ρ − =1 P ) 1 ( 1 1 1 λ ρ ρ λ μ λ = = − = − = w L

( )

1 ) 1 (

2 2

ρ ρ λ μ μ λ ρ μ λ μ − = − = − −

Q

L

Dr. Mesut Güneş

Chapter 8. Queueing Models 44

( )

) 1 ( ρ μ ρ λ μ μ λ − = − =

Q

w

SLIDE 45

Computer Science, Informatik 4 Communication and Distributed Systems

M/M/1 Queues M/M/1 Queues Single-chair unisex hair-styling shop Single chair unisex hair styling shop

Interarrival and service times are exponentially distributed
λ = 2 customers/hour and µ = 3 customers/hour

= = 3 2 μ λ ρ Customers 2 2 3 2 = − = − = λ μ λ L ⎟ ⎞ ⎜ ⎛ = − =

1

2 2 1 3 1 1 P P ρ h 2 1 1 1 hour 1 2 2 = = = λ L w = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ =

2 2 1

4 2 1 9 3 3 P P Customers 3 4 ) 2 3 ( 3 4 ) ( hour 3 3 1

2

= = = = − = − = λ λ μ

Q Q

L w w

∑

= ≥

= − = ⎟ ⎠ ⎜ ⎝

3 4 2

81 16 1 27 3 3

n n

P P Customers 2 3 2 3 4 3 ) 2 3 ( 3 ) ( = + = + = − − μ λ λ μ μ

Q Q

L L

Dr. Mesut Güneş

Chapter 8. Queueing Models 45

n

SLIDE 46

Computer Science, Informatik 4 Communication and Distributed Systems

M/M/1 Queues M/M/1 Queues

Example: M/M/1 queue with

p q service rate μ =10 customers per hour.

Consider how L and w

λ 5 6 7.2 8.64 10 ρ 0.5 0.60 0.72 0.864 1

Consider how L and w

increase as arrival rate, λ, increases from 5 to 8.64 by increments of 20%

L 1.0 1.50 2.57 6.350 ∞ w 0.2 0.25 0.36 0.730 ∞

increments of 20%

If λ/μ ≥ 1, waiting lines tend to

continually grow in length

14 16 18 20 L w

continually grow in length

Increase in average system

8 10 12 14 mber of Customers

time (w) and average number in system (L) is highly nonlinear as a function of ρ.

2 4 6 Num

Dr. Mesut Güneş

Chapter 8. Queueing Models 46

0.5 0.6 0.7 0.8 0.9 1 rho

SLIDE 47

Computer Science, Informatik 4 Communication and Distributed Systems

Effect of Utilization and Service Variability Effect of Utilization and Service Variability

For almost all queues, if lines are too long, they can be reduced by

For almost all queues, if lines are too long, they can be reduced by decreasing server utilization (ρ) or by decreasing the service time variability (σ2). A f th i bilit f di t ib ti

A measure of the variability of a distribution,
coefficient of variation (cv):

) (X V

[ ]2

2

) ( ) ( ) ( X E X V cv =

The larger cv is, the more variable is the distribution relative to its

expected value

For exponential service times with rate µ
E(X) = 1/µ
V(X) = 1/µ2
Dr. Mesut Güneş

Chapter 8. Queueing Models 47

( ) µ cv = 1

SLIDE 48

Computer Science, Informatik 4 Communication and Distributed Systems

Effect of Utilization and Service Variability Effect of Utilization and Service Variability

Consider LQ for any M/G/1 queue:

Q

y q

+ = ) 1 (

2 2 2

L μ σ ρ ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ + ⎟ ⎟ ⎞ ⎜ ⎜ ⎛ = − = ) ( 1 ) 1 ( 2

2 2

cv LQ ρ ρ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠ ⎜ ⎝ − 2 1 ρ

L for M/M/1 LQ for M/M/1 queue

C t th M/M/1 Corrects the M/M/1 formula to account for a non-exponential service time dist’n

Dr. Mesut Güneş

Chapter 8. Queueing Models 48

SLIDE 49

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue – M/M/c Multiserver Queue – M/M/c

M/M/c/∞/∞ queue: c servers operating in parallel

M/M/c/∞/∞ queue: c servers operating in parallel

Arrival process is poisson with rate λ
Each server has an independent and identical exponential service-time

distribution with mean 1/ distribution, with mean 1/μ.

To achieve statistical equilibrium, the offered load (λ/μ) must satisfy

λ/μ < c, where λ/(cm) = ρ is the server utilization.

1

Waiting line Calling population

λ 2

Dr. Mesut Güneş

Chapter 8. Queueing Models 49

c

SLIDE 50

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue – M/M/c Multiserver Queue – M/M/c

The steady-state parameters for M/M/c

The steady state parameters for M/M/c

μ λ ρ c = λ μ μ μ λ μ λ c c c n P

c c n n

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =

− − =

∑

! 1 ! ) / (

1 1

( ) ( )

ρ ρ ρ ρ c L P P c c P c c L P

c c

≥ ∞ ⋅ − = ≥ ∞

+

) ( ) ( ) 1 ( ! ) ( ) (

1

Probability that ll

( )

λ ρ ρ ρ ρ ρ ρ L w c L P c c c P c c L = − ≥ ∞ + = − + = 1 ) ( ) 1 )( ! ( ) (

2

all servers are busy

( )

ρ ρ λ c L P LQ − ≥ ∞ ⋅ = 1 ) (

Dr. Mesut Güneş

Chapter 8. Queueing Models 50

ρ c L L

Q =

−

SLIDE 51

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue – M/M/c Multiserver Queue – M/M/c

Probability of empty system Number of customers in system Probability of empty system Number of customers in system

Dr. Mesut Güneş

Chapter 8. Queueing Models 51

SLIDE 52

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue – Common Models Multiserver Queue – Common Models

Other common multiserver queueing models

Other common multiserver queueing models

⎟ ⎞ ⎜ ⎛ + ⎟ ⎞ ⎜ ⎛ ) ( 1

2 2

cv ρ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 2 ) ( 1 1 cv LQ ρ ρ

LQ for M/M/1 queue

Corrects the M/M/1 formula

M/G/c/∞: general service times and c parallel server. The parameters can

be approximated from those of the M/M/c/∞/∞ model. f f

M/G/∞: general service times and infinite number of servers.
M/M/c/N/∞: service times are exponentially distributed at rate μ and c

servers where the total system capacity is N ≥ c customer. When an arrival

Dr. Mesut Güneş

Chapter 8. Queueing Models 52

ccurs and the system is full, that arrival is turned away.

SLIDE 53

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue – M/G/∞ Multiserver Queue – M/G/∞

M/G/∞: general service times and infinite number of servers

g

customer is its own server
service capacity far exceeds service demand
when we want to know how many servers are required so that

when we want to know how many servers are required so that customers are rarely delayed

( )

1 = =

− n

n e P

μ λ

, 1 , , !

0 =

= =

− n

e P n n e P

μ λ

K 1 = w μ = =

Q

L w μ λ

Dr. Mesut Güneş

Chapter 8. Queueing Models 53

=

Q

L μ

SLIDE 54

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue – M/G/∞ Multiserver Queue – M/G/∞

How many users can be logged in simultaneously in a computer

y gg y p system

Customers log on with rate λ = 500 per hour
Stay connected in average for 1/µ =180 minutes = 3 hours

y g µ

For planning purposes it is pretended that the simultaneous logged in

users is infinite

Expected number of simultaneous users L

p

1500 3 500 = ⋅ = = μ λ L

To ensure providing adequate capacity 95% of the time, the number of

parallel users c has to be restricted

μ

parallel users c has to be restricted

∑ ∑

= − =

≥ = = ≤ ∞

c n n c n n

n e P c L P

1500

95 . ! ) 1500 ( ) ) ( (

Dr. Mesut Güneş

Chapter 8. Queueing Models 54

The capacity c =1564 simultaneous users satisfies this requirement

SLIDE 55

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue with Limited Capacity Multiserver Queue with Limited Capacity

M/M/c/N/∞: service times are exponentially distributed at rate μ and

p y μ 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
Effective arrival rate λ is defined as the mean number of arrivals per
Effective arrival rate λe is defined as the mean number of arrivals per

time unit who enter and remain in the system

a a P

c N c n c n

ρ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + =

− −

∑ ∑

! ! 1

1

λ a =

P c c a P c n

c N N N n c n

ρ = ⎥ ⎦ ⎢ ⎣

− = + =

∑ ∑

! ! !

1 1

μ λ ρ μ c =

( )

P c N c a P L

N e c N c N c Q

λ λ ρ ρ ρ ρ ρ − = − − − − − =

− −

) 1 ( ) 1 ( ) ( 1 ) 1 ( !

μ

w w L w

e Q Q

λ + = = 1

(1 - PN) probability that a customer will find a space and be able to

Dr. Mesut Güneş

Chapter 8. Queueing Models 55

w L w w

e Q

λ μ = + =

space and be able to enter the system

SLIDE 56

Computer Science, Informatik 4 Communication and Distributed Systems

Multiserver Queue with Limited Capacity Multiserver Queue with Limited Capacity

Single-chair unisex hair-styling shop (again!)

Space only for 3 customers:
ne in service and two waiting
First computer P0
Queue time

246 . 114 28 = = =

Q Q

L w λ

System time, time in shop

415 . 3 2 3 2 3 2 1 1

3 2 1

= ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + =

∑

= − n n

P

114

e

λ

66 1

P(system is full)
Expected number of

( )

123 . 65 8 1 ! 1

2 3 3 2 3

= = = = P P P

N

579 . 114 66 1 = = + = μ

Q

w w

Average of the queue
Expected number of

customers in shop

65 1 ! 1

431 . =

Q

L 015 1 66 = = = w L λ

Effective arrival rate
Probability of busy shop

754 . 1 114 8 1 2 = = ⎟ ⎞ ⎜ ⎛ − = λ

015 . 1 65 w L

e

λ

Dr. Mesut Güneş

Chapter 8. Queueing Models 56

754 . 1 65 65 1 2 ⎟ ⎠ ⎜ ⎝

e

λ

585 . 1 = = − μ λe P

SLIDE 57

Computer Science, Informatik 4 Communication and Distributed Systems

Steady-state Behavior of Finite-Population Models Steady state Behavior of Finite Population Models

Dr. Mesut Güneş

Chapter 8. Queueing Models 57

SLIDE 58

Computer Science, Informatik 4 Communication and Distributed Systems

Steady-State Behavior of Finite-Population Models Steady-State Behavior of Finite-Population Models

In practical problems calling population is finite
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.

Consider a finite-calling population model with K customers (M/M/c/K/K)

Th ti b t th d f i i it d th t ll f i i

The time between the end of one service visit and the next call for service is

exponentially distributed with mean = 1/λ.

Service times are also exponentially distributed with mean 1/µ.
c parallel servers and system capacity is K

c parallel servers and system capacity is K.

K C

1

Waiting line with capacity K

2

K Customers Mean runtime

1/λ 2

Dr. Mesut Güneş

Chapter 8. Queueing Models 58

c

SLIDE 59

Computer Science, Informatik 4 Communication and Distributed Systems

Steady-State Behavior of Finite-Population Models Steady-State Behavior of Finite-Population Models

Some of the steady-state probabilities of M/M/c/K/K :

So e o t e steady state p obab t es o / /c/ / μ λ μ λ c c n K K n K P

K n c n c n

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∑ ∑

− − −

! )! ( !

1 1

μ λ μ μ c n P n K P c c n K n

n c n n

⎪ ⎪ ⎨ ⎧ − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎢ ⎣ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝

= =

1 ,..., 1 , , ! )! ( μ λ μ K c c n c c n K K P

n c n n

⎪ ⎪ ⎩ ⎪ ⎨ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎠ ⎝ ⎠ ⎝ =

−

,... 1 , , ! )! ( ! μ λ ρ λ c L w nP L

e e K n n

= = =∑

=

, / ,

∑

− =

K n e

P n K ) ( λ λ

service) xiting entering/e (or queue to customers

f

rate arrival effective run long the is where

e

λ

Dr. Mesut Güneş

Chapter 8. Queueing Models 59

∑

= n n e

) (

SLIDE 60

Computer Science, Informatik 4 Communication and Distributed Systems

Steady-State Behavior of Finite-Population Models Steady-State Behavior of Finite-Population Models Example: two workers who are responsible for 10 milling p p g machines.

Machines run on the average for 20 minutes, then require an

average 5-minute service period both times exponentially average 5 minute service period, both times exponentially distributed: λ = 1/20 and μ = 1/5.

All of the performance measures depend on P0:

1

⎤ ⎡

−

065 . 20 5 2 ! 2 )! 10 ( ! 10 20 5 10

1 10 2 2 1 2

= ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

= − − =

∑ ∑

n n n n n

n n P

Then, we can obtain the other Pn, and can compute the expected

number of machines in system:

10

The average number of running machines:

machines 17 . 3 = =∑

= n n

nP L

Dr. Mesut Güneş

Chapter 8. Queueing Models 60

The average number of running machines:

machines 83 . 6 17 . 3 10 = − = − L K

SLIDE 61

Computer Science, Informatik 4 Communication and Distributed Systems

Networks of Queues Networks of Queues

Dr. Mesut Güneş

Chapter 8. Queueing Models 61

SLIDE 62

Computer Science, Informatik 4 Communication and Distributed Systems

Networks of Queues Networks of Queues

Many systems are naturally modeled as networks of single queues

y y y g q

Customers departing from one queue may be routed to another
The following results assume a stable system with infinite calling
The following results assume a stable system with infinite calling

population and no limit on system capacity:

Provided that no customers are created or destroyed in the queue, then

the departure rate out of a queue is the same as the arrival rate into the the departure rate out of a queue is the same as the arrival rate into the queue, over the long run.

If customers arrive to queue i at rate λi, and a fraction 0 ≤ pij ≤ 1 of them

are routed to queue j upon departure, then the arrival rate from queue i

Dr. Mesut Güneş

Chapter 8. Queueing Models 62

q j p p , q to queue j is λi pij over the long run.

SLIDE 63

Computer Science, Informatik 4 Communication and Distributed Systems

Networks of Queues Networks of Queues

The overall arrival rate into queue j:

e o e a a a ate to queue j

∑

+ =

i ij i j j

p a

all

λ λ

Arrival rate from outside the network Sum of arrival rates from other queues in network

If queue j has cj < ∞ parallel servers, each working at rate μj, then the

long-run utilization of each server is: (where ρj < 1 for stable queue).

j

λ

If arrivals from outside the network form a Poisson process with rate aj

for each queue j and if there are c identical servers delivering

j j j j

c μ ρ = for each queue j, and if there are cj identical servers delivering exponentially distributed service times with mean 1/μj, then, in steady state, queue j behaves likes an M/M/cj queue with arrival rate

∑

Dr. Mesut Güneş

Chapter 8. Queueing Models 63

∑

+ =

i ij i j j

p a

all

λ λ

SLIDE 64

Computer Science, Informatik 4 Communication and Distributed Systems

Network of Queues Network of Queues

C t

80 c = ∞ 0.4 Cust

Customer Population

80 c = 1 0.6 Cust. hour

Discount store example:
Suppose customers arrive at the rate 80 per hour and

40% choose self-service. 40% choose self service.

Hence:
Arrival rate to service center 1 is λ1 = 80(0.4) = 32 per hour
Arrival rate to service center 2 is λ2 = 80(0.6) = 48 per hour.

2

( ) p

c2 = 3 clerks and μ2 = 20 customers per hour.
The long-run utilization of the clerks is:

ρ2 = 48/(3*20) = 0.8 ρ2 ( )

All customers must see the cashier at service center 3,

the overall rate to service center 3 is λ3 = λ1 + λ2 = 80 per hour.

Dr. Mesut Güneş

Chapter 8. Queueing Models 64

If μ3 = 90 per hour, then the utilization of the cashier is:

ρ3 = 80/90 = 0.89

SLIDE 65

Computer Science, Informatik 4 Communication and Distributed Systems

Summary Summary

Introduced basic concepts of queueing models.
duced bas c co cep s o queue g
de s
Showed how simulation, and some times mathematical analysis, can be

used to estimate the performance measures of a system.

Commonly used performance measures: L L

w w ρ and λ

Commonly used performance measures: L, LQ, w, wQ, ρ, and λe.
When simulating any system that evolves over time, analyst must decide

whether to study transient behavior or steady-state behavior.

Si l f l i t f th t d t t b h i f

Simple formulas exist for the steady-state behavior of some queues.
Simple models can be solved mathematically, and can be useful in providing

a rough estimate of a performance measure.

Dr. Mesut Güneş

Chapter 8. Queueing Models 65