kendall's notation

1

Copyright 2002, Sanjay K. Bose 1

Basic Queueing Theory

M/M/-/- Type Queues


Kendall’s Notation for Queues

A/B/C/D/E

Shorthand notation where A, B, C, D, E describe the queue

Applicable to a large number of simple queueing scenarios

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Kendall’s Notation for Queues A/B/C/D/E

A Inter-arrival time distribution

B Service time distribution

C Number of servers

D Maximum number of jobs that can be there in thesystem (waiting and in service)

Default ∞ for infinite number of waiting positions

E Queueing Discipline (FCFS, LCFS, SIRO etc.)

Default is FCFS

M exponentialD deterministicEk Erlangian (order k)G general

M/M/1 or M/M/1/∞ Single server queue with Poisson arrivals,exponentially distributed service times and infinite number ofwaiting positions


Little’s Result

N=λW (2.9)

Nq=λWq (2.10)

Result holds in general for virtually all types of queueingsituations where

λ = Mean arrival rate of jobs that actually enter the system

Jobs blocked and refused entry into the system will not becounted in λ

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Time t

Num

ber

of c

usto

mer

s(a

rriv

als/

depa

rtur

es)

α(t), Arrivals in (0,t)

β(t), Departures in (0,t)

N(t

increments byone

Graphical Illustration/Verification of Little's Result

Little’s Result


Little’s Result

Consider the time interval (0,t) where t is large, i.e. t→∞

∫ −t

dttt0

)]()([ βαArea(t) = area between α(t) and β(t) at time t =

Average Time W spent in system =)(

)(lim

t

tAreat α∞→

Average Number N in system =)(

)()(lim

)(lim

t

tArea

t

t

t

tAreatt α

α∞→∞→ =

Therefore, N= λWt

tt

)(lim

αλ ∞→=Since,

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The PASTAProperty

“Poisson Arrivals See Time Averages”

pk(t) =P{system is in state k at time t }

qk(t) =P{an arrival at time t finds the system in state k}

N(t) be the actual number in the system at time t

A(t, t+∆t) be the event of an arrival in the time interval (t, t+∆t)

{ }{ } { }

{ } )(,(

)(|)(|,(lim

,(|)(lim)(

0

0

tptttAP

ktNPktNtttAP

tttAktNPtq

kt

tk

=∆+

==∆+=

∆+==

→∆

→∆Then

because P{A(t, t+∆t)|N(t) = k} = P{A(t, t+∆t)}


Equilibrium Solutions for M/M/-/- Queues

Method 1: Obtain the differential-difference equations as in Section 1.2or Section 2.2. Solve these under equilibrium conditions along with thenormalization condition.

Method 2: Directly write the flow balance equations for proper choiceof closed boundaries as illustrated in Section 2.2 and solve these alongwith the normalization condition.

Method 3: Identify the parameters of the birth-death Markov chain forthe queue and directly use equations (2.7) and (2.8) as given in Section2.2.

In the following, we have used this approach

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M/M/1 (or M/M/1/∞∞ ) Queue

,.......3,2,1

00

==

==

∀=

k

k

k

k

k

µ

µ

λλk

k

k ppp ρµλ

00 =

=

)1(0 ρ−=p

For ρ <1

ρρ

ρρ−

=−== ∑∑∞

=

∞

= 1)1(

00 i

i

ii iipN

)1(

1

ρµλ −==

NW

UsingLittle’sResult

)1(

1

ρµρ

µ −=−= WWq

)1(

2

ρρ

λ−

== qq WNUsingLittle’sResult


M/M/1/∞∞ Queue with Discouraged Arrivals

,.......3,2,1

00

1

==

==

∀+

=

k

k

kk

k

k

µ

µ

λλ

For ρ = λ/µ < ∞

∏−

=

=

+=

1

000 !

1

)1(

k

i

k

k kp

ipp

µλ

µλ

)exp(0 µλ

−=p

(2.14)

(2.15)

µλ

== ∑∞

=0kkkpN ∑

∞

=

−−==

0

)exp(1k

kkeff pµλ

µλλ

Effective ArrivalRate

−−

==)exp(12

µλ

µ

λλeff

NW Little’s

Result

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M/M/1/∞ Queue with Discouraged Arrivals

In this case, PASTA is not applicable as the overall arrival process isnot Poisson

πr = P{arriving customer sees r in system (before joining the system)}∆E be the event of an arrival in (t, t+∆t)Ei is the event of the system being in state i

∑∞

=

∆

∆=

∆∆

=∆=

0

}|{}{

}|{}{

}{

}|{}{}|{

iii

rrrrrr

EEPEP

EEPEP

EP

EEPEPEEPπ

!

1}{ /

iepEP

i

ii

== −

µλµλ

1}|{

+∆

=∆i

tEEP i

λ

−+

= −

−+

µλ

µλ

µλ

π/

/1

1)!1(

1

e

e

r

r

r ∑∞

=−−

=+

=0

/2 )1(

1

kk

e

kW µλµ

λπ

µ


M/M/m/∞∞ Queue (m servers, infinite number of waitingpositions)

kk ∀= λλmkm

mkkk

≥=

−≤≤=

µ

µµ )1(0

For ρ = λ/µ < m

mkformm

p

mkfork

pp

mk

k

k

k

>=

≤=

−!

!

0

0

ρ

ρ

11

00 )(!!

−−

=

−+= ∑

m

k

mk

mm

m

kp

ρρρ

(2.16)

(2.17)

∑∞

= −==

mk

m

k mm

mpmCp

)(!),( 0 ρ

ρρ (2.18)P{queueing} =

Erlang’sC-Formula

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M/M/m/m Queue (m server loss system, no waiting)

)(0 ConditionLossorBlockingotherwise

mkk

=

<= λλ

otherwise

mkkk

0

0

=≤≤= µµ

otherwise

mkfork

ppk

k

0!0

=

≤=ρ

∑=

=m

k

k

k

p

0

0

!

1

ρ

(2.19)

(2.20)

For

∞<=µλ

ρ


M/M/m/m Queue (m server loss system, no waiting)

Simple model for a telephone exchange where a line is given onlyif one is available; otherwise the call is lost

Blocking Probability B(m,ρ) = P{an arrival finds all servers busyand leaves without service}

!),( 0 m

pmBmρ

ρ = Erlang’s B-Formula (2.21)

1),0( =ρB

m

mBm

mB

mB),1(

1

),1(

),(ρρ

ρρ

ρ−

+

−

= (2.22)

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M/M/1/K Queue (single server queue with K-1 waitingpositions)

)(0 ConditionLossorBlockingotherwise

Kkk

=

<= λλ

otherwise

Kkk

0=

≤= µµ

otherwise

Kkforpp kk

00

=≤= ρ

)1(

)1(10 +−

−=

Kp

ρρ

(2.23)

(2.24)

For

∞<=µλ

ρ


M/M/1/-/K Queue (single server, infinite number ofwaiting positions, finite customer population K)

otherwise

Kkk

0=

≤= µµ

)(0

)(

ConditionLossorBlockingotherwise

KkkKk

=

<−= λλ

(2.25)

(2.26)

)!(

!0 kK

Kpp k

k −= ρ

∑= −

=K

k

k

kK

Kp

0

0

)!(

!

1

ρ

k=1,.….,KFor

∞<=µλ

ρ

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Delay Analysis for a FCFS M/M/1/∞∞ Queue(Section 2.6.1)

Q: Queueing Delay (not counting service time for an arrival

pdf fQ(t), cdf FQ(t), LQ(s) = LT(fQ(t)}

W: Total Delay ( waiting time and service time) for an arrival

pdf fW(t), cdf FW(t), LW(s) = LT(fW(t)}

W=Q+T

T: Service Timet

T etf µµ −=)( tT etF µ−=)(

)()(

µµ+

=s

sLT

Knowing the distribution of either W or Q, the distribution of theother may be found

SinceQ ⊥ T

][*)()()()(

)( tQWQW etftfsL

ssL µµ

µµ −=+

= (2.30)


For a particular arrival of interest -

FQ(t) = P{queueing delay ≤ t} = P{queueing time=0} + [Σn≥1 P{queueing time≤ t | arrival found

n jobs in system}]pn

)1()1()1()1(

)!1(

)()1()1(

)!1(

)()1()1()(

)1()1(

0

1

1

0

1 0

1

ρµρµ

µ

µ

ρρµρρρ

ρµµρρρ

µµρρρ

−−−−

∞

=

−−

∞

= =

−−

−+−=−+−=

−−+−=

−−+−=

∫

∑∫

∑ ∫

txt

n

nx

t

n

t

x

xn

nQ

edxe

dxn

xe

dxen

xtF

Erlang-n distribution forsum of n exponential r.v.s

(2.31)

)1()1()1)(()(

)( ρµρλρδ −−−+−== tQQ et

dt

tdFtf (2.32)

tt

xxttW edxeeetf )(

0

))(1( )()1()1()( λµµρµµ λµµρλµρ −−−−−−− −=−+−= ∫

10


Departure ofcustomer of

interest

Arrival ofcustomerof interest

Arrivals coming while thecustomer of interest is in the

system

Time spent in system by thecustomer of interest

Arrival/Departure of Customer/Job ofInterest from a FCFS M/M/1Queue

• PASTA applicable to thisqueue

• N and NQ seen by an arrivalsame as the time-averagedvalues

Let

N* = Number in the system that a job will see left behind when it departs

pn*=P{N*=n} for N*=0, 1,....,∞


Departure ofcustomer of

interest

Arrival ofcustomerof interest

Arrivals coming while thecustomer of interest is in the

system

Time spent in system by thecustomer of interest

For a FCFS queue, number leftbehind by a job will be equalto the number arriving while itis in the system.

∫

∑ ∫∑∞

−−

∞

=

−∞

=

∞

=

−==

==

0

)1(

0 00

**

)()(

)(!

)()(

zLdttfe

dttfen

tzpzzG

WWzt

nW

t

t

nn

nn

n

λλ

λ

λ

λ

Wds

sdL

dz

zdGNE

s

W

z

λλ =−==== 01

** )()(}{

(2.36)

(2.37)

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An important general observation can also be made along the linesof Eq. (2.36).

Consider the number arriving from a Poisson process with rate λin a random time interval T where LT(s)=LT{fT(t)}. The generatingfunction G(z) of this will be given by

)()( zLzG T λλ −=

and the mean number will be E{N}= λ E{T}

This result will be found to be useful in various places in oursubsequent analysis.


Delay Analysis for the FCFS M/M/m/∝∝ Queue(Section 2.6.2)

Using an approach similar to that used for the M/M/1 queue, we obtainthe following

)()!1(

)()(!

1)()(

00 tu

m

ept

mm

mptf

tmmm

Q

−+

−

−=−− ρµρµ

δρ

ρ

−−−−

−

−−=

−−−−

)1()!1(

][

)(!1)(

)(0

0 ρρµ

µρ

ρ µρµµ

mm

eepe

mm

mptf

ttmmt

m

W

(2.34)

(2.35)

See Section 2.6.2 for the details and the intermediate steps

kendall's notation

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