EMgt 356Simulation

Queuing Theory (Chapter 5)Analytical ModelsAnalytical Models

The Queueing SystemArrivalsArrivals

S ( )Server(s)

Departures

Queueing SystemServed Customers

Queueing System

CustomersQueue

Service f ilit

SSS

CCC

C C C C C C CfacilityS

SCC

Served Customers

Queueing Applications

• Fast Food • Supermarkets

• Airports

C i ti

• Post Office

T ffi• Communications

• Equipment Selection

• Traffic

• Amusement ParksEquipment Selection

• Emergency Response

Amusement Parks

• Parking Lotsg y p g

Queueing Notation and Relationships

• Characteristics– arrival pattern

• Performance Measures

– service pattern– queue discipline

– utilization– cycle time

WIP– capacity

’

– WIP

• Kendall’s Notation

Kendall’s Notation

Inter-arrival Service Time # of SystemA / B / c / K

TimeDistribution

Distribution Servers Capacity

•M = Exponential•G = General (typically implies only mean and variance may be known)

Dropped if infiniteG General (typically implies only mean and variance may be known)

•D = Deterministic (variance is zero)

Random variables

• Roughly speaking these are quantities that do not assume the same value all the time.

• Often characterized by their distributionOften characterized by their distribution and density or mass functions

• Important characteristics: Mean and• Important characteristics: Mean and VarianceDi t ib ti i di t f il d• Distribution indicates a family and can be defined by the distribution function

Continuous and DiscreteContinuous and Discrete Random Variables

• Continuous random variables can assume ALL POSSIBLE values in the range over which it is defined. Examples: service time in a queue, inter-arrival time, time between successive failuressuccessive failures.

• Discrete Random Variables can assume a countable number of DISTINCT valuescountable number of DISTINCT values. Examples: number of thunderstorms in a month, number of patients in a doctor’s clinic. o , u be o pa e s a doc o s c c

Continuous and DiscreteContinuous and Discrete (Contd.)

• Continuous Random Variables: Examples: Exponential, uniform, p p , ,Erlang, gamma, normal.

• Discrete Random variables:E l P i bi i l B lliExamples: Poisson, binomial, Bernoulli, multinomial, generic.

Distributions for continuousDistributions for continuous Random variables

• Exponential Distribution– Although it sometimes provides a good fit for inter-

arrival times, this is much less true for service titimes.

– Waterfall histogram (pdf)• Uniform distribution (rectangle histogram)Uniform distribution (rectangle histogram)• Erlang Distribution (mixture of two or more exponentials) • Normal Distribution (Bell curve histogram)

Exponential Distribution:Exponential Distribution:Waterfall Histogram

Exponential distribution

• Defined by a single parameter, e.g., λ• If the mean is 1/λIf the mean is 1/λ• The variance is 1/λ2.

Uniform distribution

• Unif (a,b)

• Mean = a+b2

• Variance = (b−a)212

Normal distribution

• Characterized by the famous Bell curve• SymmetricSymmetric• Defined by its two parameters μ and σ

Th i l t 2• The mean is equal to μ2

• The variance is equal to σ2

Some queuing randomSome queuing random variables

• Service time: time for one service• Inter-arrival time: time betweenInter arrival time: time between

successive arrivals

Performance Measures

• Average number in the queue: Lq

• Average wait in the queue: Wqg q q

• Average number in the system: L or Lsyst

• Average wait in the system: W or Wsystg y syst

• Proportion of time server is busy: ρ

Some notation andSome notation and relationships

• λ = Arrival Rate• μ = Service Rateμ Service Rate• λ = 1/E[Inter-arrival time]

1/E[ i ti ]• μ = 1/E[service time]

• Also note: ρ = λAlso note: ρ = μ

Little’s Law

Little’s Law • relates the average wait to the averagerelates the average wait to the average

length)• Works for both system and queue• Works for both system and queue• L=W and Lq=Wq

Important Relation

• We will need the following important relationship:p

• W = W + E[Service Time] (1)• W = Wq + E[Service Time] …..(1)i.e.,

1W =Wq +1μμ

M/M/1 queue

Lq = E[# in queue] or average number in the queueq

qL

L ρ2Lq =ρ1−ρ

All other performanceAll other performance measures

All other performance measures can be obtained by using Little’scan be obtained by using Little s rule and relationship (1).

Some more notation

• C2 = coefficient of variation = variance/(mean)2 variance/(mean)

c = coefficient of variation of service ti

C2stime

= coefficient of variation of inter-arrival C2atime

Exponential distribution forExponential distribution for inter-arrival and service times

• Inter-arrival time’s mean is 1/λ• Inter-arrival time’s variance is 1/λ2Inter arrival time s variance is 1/λ• Inter-arrival time’s coeff. of variation is 1

S i ti ’ i 1/• Service time’s mean is 1/μ• Service time’s variance is 1/μ2

• Service time’s coeff. of variation is 1

M/G/1 queue

Lλ2σ2s+ρ

2

Lq = s+ρ2(1−ρ)

G/G/1 queue

ρ2(1+C2)(C2+ρ2C2)Lq

ρ (1+Cs )(Ca+ρ Cs )2(1 )(1+ 2C2)q 2(1−ρ)(1+ρ2C2

s )