On a single server queue with fixed accumulation level, state dependent service, and semi Markov modulated input flow

Internat. J. Math. & Math. VOL. 15 NO. 3 (1992) 593-600

ON A SINGLE-SERVER

QUEUE

WITH

FIXED ACCUMULATION

LEVEL, STATE

DEPENDENT

SERVICE,

AND SEMI-MARKOV

MODULATEO INPUT FLOW

JEWGENI H. DSHALALOW and

GARYRUSSELL

Department

of Applied

Mataematics

FloridaInstituteofTechnology

Melbourne,

FL

32901,

USA

(Received

July

29,1991)

ABSTRACT.

The authors study the queueingprocess inasingle-server queueingsystemwith statedependent service and withthe input modulated byasemi-Markovprocessembedded in

thequeueingprocess.

It

isalso assumed that theservercapacityis r

>

1 and thatany service

act will not begin untilthe queue accumulates at least runits.

In

this

model, therefore,

idle periods alsodependuponthe queuelength.

The authors establish an ergodicity criterion for the queueing process and evaluate explicitlyitsstationary distribution and other characteristics of thesystem, suchas themean service cycle, intensity of the system, intensity of the input stream, distribution of the idle period,and themeanbusyperiod. Variousspecialcasesaretreated.

KEY

WORDS

AND PHRASES.

Queueingprocess, semi-Markov process, semi-regenerative pro-cess, embedded Markov chain, semi-Markov modulated Poisson process, equilibrium, conti-nuoustimeparameterprocess,servicecycle,idleperiod, busyperiod, intensity of theprocess.

1991 AMS

SUBJECT CLASSIFICATION CODES.

Primary60KI0, 60K15, 60K25, Secondary 90B22,90B25.

i.

INTRODUCTION.

Many

standard queueing systems operate on the assumption that input and service

parameters are independent of the state of the system; to assume otherwise is frequently

regarded

astoounwieldy.

In

this articleweproposeaclassofqueueingsystems whichcanbe analyzed rather

thoroughly

even

though

theinput andserviceparametersarestate

dependent.

We

add the provision that the service is delayed until the service batch size reaches the

server’s

capacity.

We

will show that such a class of systems readily avails itself to the appropriateanalyticaltechniques.

However,

for the class of systems under consideration, we assume that r

(_>

1)

units are necessary forservice, and if fewerare availabletheserver waitsfor moreunits to arrive. The different possible values of r,aswellasthe formation of theidleperiod

(which

in our casedoes not necessarily end with the arrival ofthe next

unit),

allows our system to encompass more situations. Applicationsin transportationseemprobable.

Otherauthors, suchas

Neuts,

Sumita, and Takahashi

(see

[4]),

have studiedaPoisson processmodulated byaMarkovprocess.

In

thisarticleweconsiderthemoregeneralcaseofa Poisson process modulated bya semi-Markov process.

We

alsoassumethe service process is

statedependent. Afteraformaldescriptionwestudytheembeddedqueueing process construc-tedoverthe sequence ofinstantsofservicecompletion

(under

norestriction to service time

dis-tributions).

We

extend the result for the continuous timeparameter process by using tech-niques appropriate forsemi-regenerativeprocesses.

We

establishanecessary andsufficient cri-terionforergodicity and giveexplicit formulas for the limiting distributions of the processes.

We

derivethemean servicecycle, intensity ofthe system,intensity of the inputstream,

distri-bution oftheidleperiod, and the mean busyperiod. Examplesare presented throughout the

paper.

2.

FORMAL DESCRIPTION OF THE SYSTEM

Let

{ft,

,

(P=)feE, Q(t)

t>_

0}

E

{0,1,...}

denote thenumber of units in a

single-serverqueueingsystemattimet,let

{T.

nE

N,

To

0,}

be the sequence ofsuccessive instantsofservicecompletion,let

Q.

Q(T.

+

0),

let

C(-)

be the countingmeasureassociated with the point process

{T.},

and let

(t)=

Q(Tc(t)+

0),

>_

O. Then theinput is a Poisson process modu2atedby

{(t)}

due toadefinition to follow.

Let

(t)

bean integer-valuedjump process

(with

successivejumps at

T,

nE

N,

noting that 0 is theincrementofajump inthecaseof

Q.-1

Q.)

and let

{T;

k

N}

be a non-sta-tionaryorderly Poissonpoint process withitsintensity function

A(t).

Thenwecall thedoubly stochastic Poisson point process with intensity

A((t))

the Poisson process modulated by the jmp process

{(t)}

and it is denoted by

{,}.

Let

Ne(

denote the associated counting measure.

[For

aformal definition of modulated processesseeDshalalow

[3].]

Ifthe queuelength

Q.

isatleast r,thenattime

T. +

0 the server takesabatch ofunits

ofsize rfrom the queue (according to the

FIFO discipline)

and servesit fora random time

r.

+1. Otherwise, theserveridles until the queue

length

Q(t)

first reaches levelrafter

T.

and

then it begins to process a groupof runits taken fromthe waiting roomof infinite capacity

(again,

according to the

FIFO

discipline) with actual service time again equal to

.+

1.

In

bothcases we assume that

.

₊ has aprobabilitydistribution function

B.

{B0,

B1

,...}, Bi

beinganarbitrarydistribution function with finitemean bi.

3.

EMBEDDED

PROCESS

{Q.}.

Let

V.

Ne(a.).

Thentheprocess

{Q.}

isdefinedrecursively by

Q.+I=(Q.-r)+

+

V.+,

(3.1),

QUEUEING PROCESS IN A SINGLE-SERVER QUEUEING SYSTEM 595

Q,

;n6

No}

--

E

is ahomogeneousMarkovchainwith transitionprobability matrix

T

(Po)"

Let

A,(z)

denote the generatingfunction ofith row of matrix T.

From

A,(z)=

E’[z

Q*]

and

(3.1)

weobtain

Ai(z)

z(’-")+

l,(A,

A,z),

e

E

(3.2),

where

(8), Re(8)>0,

is the Laplace-Stieltjes transform of the probability distribution function

B

and

A,

A(i).

Foranalyticaladvantageand with verylittle sacrifice ofgenerality we drop the modulation and service control when the queue length exceeds a fixed

(perhaps

very

large)

level

N. In

otherwords,weassumethat

B,(z)

S(z),

fl,(O)=

(0),

b,

b,A,

A,

i>N

(AS).

Withoutloss ofgeneralitywealsoassumethatN

>

r-1.

Given assumption

(AS),

we can show that the transition probability matrix

T

is

reducedtoaform ofthe

A,.N-matrix

introducedand studiedin

[1].

Accordingtotheorem A.1

(see

Appendix),the Markov chain

{Q.}

isrecurrent-positive if andonlyif

z

Ai(z) <

oo, 0,1

..,N,

lira

z--,lzeB(O,I)

and

(3.3)

"---:(A

Az)

<

r

(3.4).

zl:zB(O,

Condition

(3.3)

isobviouslymet andcondition

(3.4)

isequivalentto

p Ab

<

r

(3.5).

Given that p

<

r, the Mkov chin

{Q}

is

ergc.

Let

P

(p=

;x

E)

denote the

invit probability meure ofoperator

T

d let

P(z)

be the generatingfunctionof vector P. Nowweformulate the

mMn

result ofthis section.

THEOREM 1. The embedded queueing process

Q,}

is ergodic if and only if p

<

r. Underthiscondition,

P(z)

satisfiesthe equation

o

(Po,’",PN)

lz)

z

(A- lz)

(3.6a),

with

A(z)

determinedby

(3.2).

Probabilities Po, ...,PNform the uniquesolutiontothe follow-ing system oflinearequations:

EN=op,-z{A,(z)-

z

}

0, k 0,...,

ko

1,s

1,...,S

(3.6b),

E,N=oP,

tp,-

p

+

(r- i)+1

r-p

(3.6c),

where p

b,

z

aretherootsof zN+ zN+

-/(

z)in

the region

B(0,1)\{I}

with their

multiplicities

ko

such that s

k

N.

PROOF.

Formula

(3.6a)

follows from

P(z)

E

,EP,

A(z)

and

(3.2).

Itis easyto modify formula

(3.6a)

into

=N+,pz_(N+I)

=oP{A(z)

z

i}

(3.6d).

zN

+

zN

+

-,(_

z)

Obviously,

:=N

_+,

p,z’

(N+1) _{is analyticin}

B(0,1)

_{andcontinuous on}

OB(0,1).

_According

totheoremA.2, thefunction z z

-(A-

Az)

must haveexactlyrzeros

(counted

with their

multiplicities)

in

i’(0,1),

andallzeros onthe boundary

0B(0,1),

including the root 1, must be simple after we meet the ergodicitycondition p

<

r. Therefore, the denominator intheright handsideof

(3.6d)

hasexactly groots intheregion

(0,1)\{1}

andthis, along with

(P,1)=

1

(which

isequivalent to

(3.6c)),

yields the equationsin

(3.65)

and

(3.6c).

(3.6b)

and

(3.6c)

has another solution

p:

{p:; i-0,...,N}.

We substitute

p*

into

(3.6a)

to obtain the generating function

P*(z).

Then,

P*(z)

is analytic in

B(0,1)

and continuous on

0B(0,1).

Therefore,

P*

{p

;iG

}

G

(l

,

)-

Obviously, equations

P’(z)

,pA,(z)

and

,N__

0

{A,()-

(-

)}

P.()

,_

(- )

are equivalent. The last equation is also equivalent to P*

P’A.

Since p* satisfies

(3.6c)

it

follows that

(P’,I)=

1. Thus, the system of equations z=

zA, (z,1)=

1 has two different

solutionsin

(l

,

-II)

whichisimpossible. [’1

4.

EXAMPLES AND APPLICATIONS.

DEFINITIONS

2.

(i)

Let

/j

EJ[T],

the mean sojourn time ofthe process

{(t)}

instate {j}, and let j

(/;j

E

E)

T.

Then

PJ

is the mean service cycle

of

the system, where

P

denotes the

stationaryprobabilitydistributionvectorofthe embedded queueingprocess

(ii)

Let

,=(x;zEE)

T _{and let}

p=.

_{be the HaAamard}

(entry-wise)

_{product of}

vectors/

and

.

Wecall thescalarproduct

Pp

theintensity

of

thesystem.

The concept of "intensity of the system goes back to the classical

M/G/1

system, where

Pp

reduces to p Ab. It is worth noting that the intensity of the system and the capacity of theserver

(in

our case

r)

coincide,asstated in proposition4andproved thereafter.

PROPOSITION

3. Given the equilibrium condition p

<

r, the mean service cycle satisfies thefollowingequation:

P-

b/

=opj[(b,-b)/

(r-

i)

+

(4.1).

PROOF.

Obviously,

/j

b

+

(r

j)

+/A.

The statement is now due to elementary

algebraictransformations. 13

PROPOSITION

4. Giventheergodicitycondition p

<

r, the intensity ofthe systemand and the capacity of theservercoincide.

PROOF. From

definitionof

Pp

and considerationsasinpropositi.on3itfollowsthat

Pp=

p+

N=op[(pj--p)+

(r-i)

+]

(4.2).

The statementisdue to relation

(3.6c)

andelementary algebraictransformations. 13

EXAMPLES5.

(i)

Consideraspecial caseofourmodelwithr

2, N

4 andwith

B

as anegative

ex-ponential distribution with parameter

}.

However,

weretain all otherassumptions aboutthe modulation andservice control having

Bo,...,B

4arbitrary.

For

thiscase we

obtain/(-

z)

(1

+

p-

pz)-

anditfollows thatthe onlyroot of the equationz

-/(A-

z)

insidethe ball

B(0,1)

is

zx

2p Thus for equation

(3.6b)

wewill be using

z

with multiplicityone

and 0 withmultiplicitythree. This willgive4 of total 5equations in the unknownsP0,.-.,P4:

E’

,=o

(z

‘-)+

,(,- ,)-

},

0

’(o)o

+

,,

(,),

+

[’()

2]

2S()

+

2,(,),

0

o;(o)po-[,(,)

+

1,- ()p

+

()p

o

QUEUEING PROCESS IN A SINGLE-SERVER

The fifth equation will be

(3.6c)

with r 2 and N 4. This system can be solved by elementary algebraic methods. The solution of this system will then be put into equation

(3.6a)

tohave thegeneratingfunction

P(z)

inanexplicit form.

(ii) By

dropping the modulation and servicecontrol and settingr 1, weimmediately

arriveatthe classical formulabyKendall establishedfor the model

M/G/1.

5. CONTINUOUS

TIME

PARAMETER

PROCESS

Q(t)}.

In

this sectionour mainobjective will be the derivation of thestationary distributionof the queueingprocess with continuous timeparameter. Priortothis,wewill be concerned with somepreliminaries.

From section 3 and definition A.4, it follows that

{fl,J,(P=)=,E,

Q(t);

t>_

0}

---,

(E,

(E))

is a semi-regenerative process with conditional regenerations at points

T.,

nE

N. Let

f,,

(P=)=,E, Q,, T,:

n 0,1,...

(E

x

R

+,

9(E

x

R

+

))

be the associated Markov renewal

processand let

Y(t)

be thecorrespondingsemi-Markovkernel. With avery mild restrictionto the probability distributionfunctions

B,

we canspecify that the elements of

Y(t)

arenot step

functionsand thus

{Q.,

T,}

is aperiodic.

By

proposition 3, themeanservice cycle

Pfl,

which jsalso themean inter-renewal timeofthe Markov renewal process, isobviouslyfinite. There-fore, followingdefinitionA.5andgiventhat p

<

r, theMarkovrenewal process isergodic.

p

It also follows that the jump process

{f,,(

)E, f(t);

_> 0}

--,

E,

definedin section

2, is theminimal semi-Markov process associated with the Markov renewal process

{Q.

,T.}

and therefore, following the definition in section2,the input process

{f,,(P)=,E,

Ne}

-

E

isaPoissonprocess modulatedbythe semi-Markov process

{f(t)}.

Let

g(t)=

(g,k(t);j,

k

e E)

be thesemi-regenerative kernel

(see

definition

A.6).

The following statement holdstrue.

LEMMA

6. The semi-regenerative kernelsatisfiesthe following equations:

(,

j,

,)(1

B())

d 0

<

j<,

<

gjk

(t)

Io

u)Aju(k--(5.1)

,,,(k-

)[1

B()],

< <

k

O,

O<k<j,

with

(;

y

e

R+

the Poissonsemi-group and

e(,n,-)

the Erlang-n probabilitydensity

func-tionwith parameter

.

PROOF. The aboveassertionfollows fromstraightforwardprobabilityarguments. [’l

Now

wearereadytoapplytheMain

Convergence

Theorem to thesemi-regenerative ker-nelinthe formofcorollary A.8, thereby arrivingat the stationary distribution of the queueing process

{Q(t)}.

THEOREM 7. Given the equilibrium condition p

<

r for the embedded process

{Q.},

the stationary distribution

(r;x e

E)

of the queueing process

{Q(t)}

exists; it is indepen-dent ofanyinitialdistribution andisexpressedinterms of thegenerating function

r(z)

of by the following formula:

where

P(z)

is the generating function of

P,

Pfl

is determined in proposition 3, and

A(z)

is defined in

(3.2).

PROOF.

Recall that the Markov renewal process

{Qn,Tn}

is ergodic if p

<

r.

By

corollary A.8 the semi-regenerative process

provided that p

<

r.

We

cansee thatthe semi-regenerative kernelis Riemann integrableover

[+.

Thus,

following corollary

A.8,

weneed tofindthe integrated semi-regenerative kernel

H

(which

is donewith routine

calculus)

and then thegeneratingfunction

h(z)

foreachrowofH.

First wefind that

zp

h(z),

i

>

r

(5.2b).

Then,

j()

1

’-

%(A-

)

A,(1

z)

,0

_<

i

<

r

(5.2c).

Formula

(5.2a)

nowfollows fromcorollary A.8,equations

(5.2b)

and

(5.2c)

andsomealgebraic transformations.

EXAMPLES

8.

(i) By

dropping themodulation of the input processweobtainfrom proposition4 that

Pfl,

--

ands(z)

rl_z)P(z).

(ii) By

using obvious probability argumentswe derive the probability density function of

anidleperiodin thesteady state:

:

0P

U- _r--1

oP

Themeanidle period inthesteadystateisthen

s

(s.3)

0p,

(iii)

Formula

(5.3a)

and theorem 7 allowustoderivethemeanbusy period

B

in

equilib-,-I

r

is the probability that the server idles.

On

the other hand, it also

rium. Clearly

0

equals_j

.

.

Thuswehave

-1

(iv)

Nowweturntothe specialcaseinexample5

(i)

applyingitsresultsfor the process with continuous time parameter.

We

use _{probabilities P0,--.,P4, substituting} them into

formulas

(5.2a)

and

(4.1)

forr 2 andN 4,thereby reducingthegeneratingfunction

r(z)

to anexplicitform.

(v)

Ifthe inputisastationaryPoissonprocess thenitsintensityis

A,

which isalso the

meannumber of arrivals per unittime.

In

thecaseofamodulatedinput processitsintensityis

nolongeratrivialfact.

We

definethe intensity of any countingmeasure

N

by the formula

where

pt(z)=

E=[N([0,])].

We

will apply ergodic theorem3.9 established inDshalalow

[3]

for

moregeneralPoisson processmodulatedbyasemi-Markov process:

:

P/P/

wherebyproposition 4.3

P

rand

P

satisfies equation

(4.1)

and thuswehave:

-

(s.3).

formula

(4.1)

combiningit withformula

(5.3b).

Then:

A,

asitshould be.

Another interesting special caseisdue to the assumption that b b andr

(observe

that we only require that the service means

bo, b,..,

are equal but do not restrict the corresponding

distributions).

Thenwehave that requalsthe reciprocal of

Pfl

b

+

_Po

o

APPENDIX

THEOREM

A.1

(Abolnikov

and Dukhovny

[1]).

Let

{Q,,}

be an irreducible aperiodic Markov chain with the transition probability matrix

A

in the

form of

a

Ar,v-matrix.

Then

{Q,}/s

recurrent-positive

if

andonly

if

lira

z

A(z)

<

,

i=O,1,...,N,

(A.la)

z-.1:zB(O,

and

lira

d-/(A

Az) <

r

(A. b).

z-,l:zeB{o, 1}

THEOREM

A.2

(Abolnikov

and Dukhovny

[1]).

Under the condition

of

(A.lb)

the

function

z

-

(A-Az)

has exactly r roots belonging to the closed unit ball

B(0,1)=

{ze

C:

< 1}.

Tho oot oth

o=d=,

0(0,X)

=

DEFINITION A.3. Let

T

be a stopping time for a stochtic process

{,, (P=)=,,

Z(t);

t0}

(E,

(E)).

The process

{Z(t)}

is sMd to have the locally strong Markov

propey

at

T

if for each bounded random viable

(:

E

d for ech

BMre

function

f:

E

R,

r 1,2,...,itholdstruethat

E’[f

o

(

oO_T

IT]

EZr[f

o

(]

P’-a.s.

on

{T < },

where is theshiftoperator.

DEFINITION A.4.

A

stochtic process

{fl,,(P=)=,e,

Z(t);t 0}

(E,

(E))

with

E 5 N

is cMledsemi-regenerative if

a)

thereisapointpreeNS

{T,}

on

R+

such that

T, (n)

d such that each

T

isastoppingtimerelative to theconicfiltering

a(Z;y

t),

b)

theprocess

{Z(t)}

h thelocMly strongMkov property at

T,,

n 1,2,...,d

c)

{Z(T,

+0),T,;

n

0,1,...}

isaMkov

renewM

preens.

DEFINITION A.5. Let

{X, ,T}

irreducible aperiic Mkov

renewM

process

with discrete state space

E. t

E=[T]

be themesojourntimeofthe Mkov

renewM

preens instate

{x}

d let

fl=

(=;x

E)T.

Suppose

that the

emded

Mkov chMn

is ergicwith stationydistributionP. WecMl

Pfl

themeaninter-renewdtime. WecMl the Mkov

renewM

process recuent-positive if its me inter-renewM time is finite.

An

ieducible

ariodic

d

recurrent-sitive

Mkov

renewM

processis cMled ergodic.

DEFINITION

A.6.

Let

{fl,,

(P=)=,E, Z(t);t 0}

(E,

(E))

beasemi-regenerative process relativetothe sequence

{T}

ofstoppingtimesd let

r(t)

P,{Z(t)

t,T

>

t},

,

e

E.

WewillcM1 thefunctionM matrix

K(t)

(K#(t)

j,k

e E)

the semi-regenerative keel.

THEOREM

A.7

(The

MMn Convergence

Whrem,

cf.

inl

[2],

p.

347).

Let

{,,

(P=).,,

z(t);

e

0}

(E,

(E))

i-#=ti

tohti=

e

sequence

{ Tn} of

stopping timesand let

K(t)

be the coespondingsemi-regenerative keel.

Suppose

that the sociated Markov renewalprocess ergodic and

at

e

semi-regenerative

process

{Z(t)}

ezistaandit is determined

from

the

formula:

rk

E,,EP,

f

_o

K,k(t)dt,

ke

E

(A.Ta).

COROLLARY

A.8.Let

H=

(hj;j,

kE

E)=

fo

K(t)dt (the

integratedsemi-regenerative

kernel),

let

h(z)

be the generating

function

of

th row

of

matri

H,

and let

x(z)

be the generating

function

of

vectorz. Then

1

h(z)

(A.Sa)

()

P--B

E

,E

V

PROOF. From

(A.Ta)

wegetanequivalent formulain matrixform, r

-.

Finally, formula

(A.Sa)

istheresult ofelementary algebraictransformations. !’!

REFERENCES

[1].

ABOLNIKOV, L.

and

DUKHOVNY, A.,

Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications,

Journ.

AppI.

Math. Stoch.Anal., 4, No. 4, 335-355, 1991.

[2].

QINLAR,

E.,

Introduction to Stochastic

Processes,

Prentice

Hall,

Englewood Cliffs,

N.J.,

1975.

I3].

DSHALALOW,

J.,

On

modulatedrandom measures.

Journ.

AppI.

Math. Stoch.

Anal.,

4, No.4, 305-312, 1991.