Monotonicity of the throughput of a closed Erlang queueing network in the number of jobs

Monotonicity of the throughput of a closed Erlang queueing

network in the number of jobs

Citation for published version (APA):

Adan, I. J. B. F., & Wal, van der, J. (1987). Monotonicity of the throughput of a closed Erlang queueing network in the number of jobs. (Memorandum COSOR; Vol. 8701). Technische Universiteit Eindhoven.

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Faculty of Mathematics and Computing Science

Memorandum COSOR 87 - 01

Monotonicity of the throughput of a closed Erlang queueing network

in the number of jobs by

Ivo Adan and Jan van der Wal

Eindhoven. The Netherlands January 1987

Monotonicity of the throughput of a closed Erlang

queueing network

in

the number of jobs.

by Iva Adan Jan van der Wal

ABSTRACT

It is shown that the throughput of a closed Erlang queueing network is nondecreasing in the number of jobs.

1. Introduction.

Recently a number of papers appeared on the problem of establishing the intuitively obvious result that the throughput in a closed queueing network is non decreasing in the number of jobs. All papers, see e.g. Robertazzi and Lazar [1985], Suri [1985] and Yao [1985]. deal with productform networks and use this productform explicitly. In Van der Wal [1985] the exponential network was treated without using the productform character of the equilibrium distribution.

In this note we show that with some modifications the proof in Van der Wal [1985] can be extended to the closed network with Erlang service times. The proof is based on the relation between continuous and discrete time Markov chains and uses mathematical induction.

The paper is organized as follows. In section 2 the model and some notations are introduced. and the main theorem is formulated. Section 3 presents some well-known results on the rela-tion between contiuous and discrete time Markov chains with rewards. The proof of the main theorem is given in section 4. To complete this proof. we use a second theorem. Section 5 con-tains the proof of this theorem.

2. The model.

We consider a closed queueing network with N single server stations. The service time is Erlang distributed, i.e. the service time at station i consists of Lj independent exponential

phases with mean Pi-I. The routing of the jobs is determined by the irreducible matrix P with

elements Pij indicating the probability that a job after its completion at queue i jumps to queue

j.

The state of the system can be characterized by the vector (

.1)

=

(k1, ... ,kN.l1 .... .l_{N) with k}i

the number of jobs in station i and Ii the phase of the job which is being served in station 1. We shall suppose that in an empty station the phase is 1. Consequently the phase is always 1 if a job has been completed. also if the station is empty afterwards. The set of all such state vec-tors with kj ~ 0, Lki = K, 1 ~ Ii ~ Lj and Ii = 1 if ki

=

0 will be denoted by S(K). So S(K)

is the set of all possible states for the queueing network containing K jobs. The queueing net-work model gives rise to an irreducible continuous time Markov chain if P is irreducible (as we already assumed).

If phase j at station i has been completed one receives a reward RCi.j). The average reward per unit time will be denoted by G(K). So

G(K)

=

r.

p(

k ,1J

r.

Pi e(ki ) RCiJi)

( !t,l.)ES(K)

where p( k,l) is the limiting probability of the network being in state ( k.l) and e(k) = 0 if k = 0 and 1 elsewhere.

The definition of throughput is more or less arbitrary. Due to the Markovian routing all throughput notions differ only by a multiplicative constant. If we define throughput as the total number of service completions in the network then we should define RCi.j) = 1 if j

=

Li and 0 otherwise. i = 1. .... N.

Now the mono tonicity can be stated as

Theorem 1. If

RCi.j) ~ 0 for all i and j then

G(K+1) ~ G(K) for all K = O. 1. 2 ... 3. Preliminaires.

Let Q be the generator of an irreducible finite state Markov chain with reward rate res) if the system is in state s. We denote this chain by (Q,r). With this continuous time chain one may associate a discrete time Markov chain with transition matrix R = I + a Q, where a

>

0 is a constant such that R is nonnegative, and with immediate reward per period ares) if the system is in state s.

The irreducible discrete Markov chain will be denoted by (R,ar). Then we have the following well known results (see Van der Wal [1985])

Lemma 1.

The equilibrium distribution of the chains (Q,r) and (R,ar) are identical. Hence the average reward per unit time for (Q,r) and the average reward per period for (R.ar) differ only from a multipicative constant a.

And

Lemma 2.

Let (R,r) be an irreducible discrete time Markov chain with rewards. and let g be the average reward per period for this chain. Let further yn be the n-period reward vector,so

n-l

yn

=

r.

Rt r t::: 0

and also

-3-Then for each state i lim n-1Vn(O = g

n-oo

An immediate Cbf1sequence of this lemma is

Lemma 3.

Let (PI,rt) and (Rz,rz) be two irreducible Markov chains with average reward per period gl and

g2 respectively. (The chains need not have the same number of states). Let V{' and V₂n be the n-period reward vector for chain 1 and 2 respectively. If for some state xin chain 1 and some state y in chain 2 we have

then

So this lemma enables us to compare different chains, in our case networks with K and K+l jobs.

4. Proof of theorem 1.

In order to prove G(K+ 1) ~ G(K) we consider the two continuous time Markov chains with rewards (QK.rK) and (QK+l.rK+1)' Here QL is the generator for the network with L jobs and rL is the reward structure defined by

Related to these continuous time chains we define the discrete time chains

with a

>

0, but sufficiently small for I + aQK and I + aQK+l to be nonnegative. Now define VR. and VR.+l to be the n-period reward vectors for the two chains. Then according to lemma 3 it suffices to show that for some state xES(K) and YES(K+l)

By induction we shall prove for all m

(2) VR'+l( li

+

1Lr ..

U

~ VR'( li.~) for all (li.l)ES(K) and r = 1.. .. ,N. where1Lr denotes the r-th unit vector: (0 ... 0.1.0 ... 0) with the 1 on place r.

Since VO

=

0 inequality (2) trivially holds for m = O. Assuming that (2) holds for m = n we prove it for m = n+1. Using recursion (1) we can express v~tf( k

+

~f

.l)

and V~+l(

k..l)

in terms of V~+1 and V~.

First V~+1( k

.l).

S(Ii - Li )

r.

Pi) V~(

k. -

~i

+

j

+

(1 - 8 (Ii - L) V ~(

k.

.l

+

~i ) }

where 8(k)

=

1 if k = 0 and 0 elsewhere. And

We rewrite this relation as

iFf

iFf

SCI; -

L;)

r.

Pi) V~+l( k

+

~r - ~i

+

~j

.l-

(Li - l)~i)

j

+

(1 -

SOi -

Lj ) V~+l(

+

~r

.l

+

~i) }

+a

Ih{ SOr - Lr)

r.

Prj V~(

+

~J

.l-

(Lr - t)~r ) j

+

(1 -

SOr -

Lr» Vi'+1(

+

~r

,l

+

~r ) }

+

(1 - a

r.

ILi e(kj ) - aILr) V~+l(

k.

+

~r

.l)

iFf

v

~tl(

k. +

~r ,l)

=

a

r.

ILi e(k;) R(U)

+

a

r.

ILi e(k) {

j

so; -

L;)

r.

Pij V~+l(

k.

+

~r - ~i

+

~j

,l-

(Li - lki) j

+

(1 -

SCI; -

L» V~+l(

+

~r

.l

+

~i ) }

+

(1 - a

r.

ILi e(kj

»

V~+l( k

+

~r

.l)

Using

5

-+

8C1r - Lr)

1:,

Pij V~+l( k

+

~j

.l-

(Lr - t)~r)

j

+

(t - 8(1r - Lr)) V~+l(

k.

+

~r

.l

+

~r )

(3) V~+l( k

+

~r - ~i

+

~j

.l-

_{(Li -} t)~i) ~ V~( k - ~i

+

~j

.l-

(Li - t)~i)

(the induction hypothese for m

=

n)

(the induction hypothese for m

=

n)

(5) (t - 0/

1:,

fJ-i e(k)) V~+l( k

+

~r

.l)

~ (1 - 0/

1:,

fJ-i e(k)) V~(

k..l)

i

(the induction hypothese for m

=

n)

we see that v~tf(

k. +

~r

.l)

~ V~+l(

k..l)

+

0/ fJ-r (t - e(kr )) { R(dr )

+

8Clr - Lr)

1:,

Pij V~+l(

k. +

~j

.l-

(Lr - 1kr) j

+

(1 - 8Clr - Lr)) V~+l(

k.

+

~r

.l

+

~r)

Clearly (2) holds for m = n+1 if we prove

Theorem 2.

If

RCi.j) ~ 0 for all i and j

then for all m and r

(6) R(r.lr)

+ (1 - SOr - Lr» V~+l( k + ~r . .1 + ~r ) ~ V~l( k

+

~r • .1)

for all (k + ~r • .1)ES(K+1) and r

=

1. .... N.

We can interpret this theorem as follows. Suppose that we have two possibilities if a phase has been completed. The first possibility is to make the jump to the next phase or queue and receive a reward. The second possibility is to do the phase all over again and receive no reward. Now theorem 2 states that we should prefer the first possibility. We shall prove theorem 2 in the next section.

So (2) holds for all m. whence. by lemma 3.

G(K+1) ~ G(K).

5. Proof of theorem 2.

For m = 0 inequality (6) holds because R(r.lr) ~ 0 (the main assumption)

Assuming that inequality (6) holds for m n we prove it for m n+1. We do this in two parts. In the first part we assume that Ir

<

Lr and in the second part we assume that Ir

=

Lr· The first part.

In case Ir

< Lr we have to prove

Using the recursion (1) we can express V-R~f( k

+

~r

of V-R+l' ~r ) and V-R~t(

k

+

~r ,.1) in terms And iFf

+

a

1:

J.li e(k) { i;o<r

SOj - L)

1:

PijV-R+l( k

+

~r - ~j

+

.

.1- (Li - 1)~I) j

+

(I -

SOj -

L) V-R+l{

k

+

~r ,.1

+

~i) }

+

(1 - a

1:

J.lj e(kj ) - a J.lr) V-R+l(

k

+

~r • .1)

7

-i¢ r

+

a

1:

Pi e(k) { i¢ r

SCli - L)

1:

PijVR+l( 1.

+

~r - ~i

+

~j,l

+

~r - (Li - 1)~i )

j

+

(1 - S(1i - L i)) VR+l(1.

+

~r,l

+

~r

+

~i) }

+

a Pr { SCIr

+

1 - Lr)

1:

PrjVR+l( 1.

+

~j , l

+

~r - (Lr - l)~r) j

+

(1 - SOr

+

1 - Lr» VR+l( 1.

+

~r , l

+

2~r ) }

+

(1 - a

1:

Pi e(k) - aPr) VR+l( k

+

~r , l

+

~r) i¢ r

If we use this relation and rewrite R(r,!r) as

apr _R(dr)

+

a

1:

Pi e(ki ) R(dr ) i¢ r

+

C1 -

a

1:

Pi e(k) - aPr) R(dr)

i¢ r

the left hand side of (7) results in

Using

i¢r

+

a

1:

Pi e(k) { i¢ r

S(1i - L i)

1:

Pij (R(dr)

+

VR+l( k

+

~r - ~i + ~j , l

+

~r - (Li - 1ki »)

j

+

C1 -

S(1i - L i» (R(dr)

+

VR+l( k

+

~r , l

+

~r

+

~i ») }

+

a Pr { R(r,!r + 1)

+

SOr

+

1 - Lr)

1:

PrjVR+l( 1.

+

~j , l

+

~r - _{(Lr - 1kr )} j

+

(1 - SOr

+

1 - Lr)) VR+l( 1.

+

~r , l

+ 2~r)

}

+

(1 - a

1:

Pi e(k) - apr) (R(dr)

+

VR+l( 1.

+

~r , l

+

~r») i¢ r

~ Vi+l( k

+

~r

-

~i

+

.1.-

(Li - l)ej)

(induction hypothesis for m = n)

(9) R(dr )

+

Vi+l( k

+

~r

,1.

+

~r

+

~i ) )

~ Vi+l(.k

+

~r

.1.

+

~i )

(induction hypothesis for m

=

n) (10) R(r.1_t

+

1)

+

8(1r

+

1 - Lr)

L

PrjVi+1( k

+

~j.1.

+

~r - (Lr - lkr)

j

+

(1 -

8CI

r

+

1 ..,;.. Lr)) Vi+l( k

+

~r

.1.

+

2~r) }

~ Vi+l( k+~r

.1.

+ ~r)

(induction hypothesis for m

=

n) (11) Redr)

+

Vi+l( k

+

~r'1.

+

~r)

~ Vi+l( k

+

~r

.1.)

(induction hypothesis for m = n) we see that

that is. (7) holds. The second part.

In case lr = Lr we have to prove

Using the recursion (1) we can express ViU( k

+

~s .1.- (Lr - 1kr) and Vi:trc.k

+

~r .1.)

in terms of Vi+l.

i¢r

80i - L)

L

PiJVl+l( k

+

~r - ~i

+

Jtj

.1.-

(Li - 1ki )

j

9

+

a I1-r

1:

PrjV~+l(.k

+

~j

,l

-

(Lr - lkr)

j

+

(1 - a 1:l1-i e(ki ) - a I1-r) V~+l(.k

+

~r ,l)

ipl r

And with fj Ij == 1 if i j and 0 elsewhere.

IpI r

SOl - L)

1:

PljVi:+l( k

+

~s - ~I

+

~j

.l-

(L_{r -} t)~r - (Li - 1ki)

j

+

(1 - SOl - Li

»

Vi:+l(.k

+

~s - (Lf - t)~r

+

~I) }

S(1 - Lr)

1:

PrjV~+l(.k

+

~s - ~r

+

~j

,l-

(L_{r -} lkr - (Lr - t)e r)

j

+

(1 - S(1 - Lr» V~+l(

+

~s

,l-

(Lr - 1kr

+

~r) }

If we split the summation over all i ;c r in summation over all i ;c r with kj

>

0 and over all

i ;c r with ki

=

0 and order terms. we see that

v

~U( k

+

~s

.l

-

eLf - 1)~r)

=

a

1:

11-1 R(i,l)

+

a

1:

11-1 { i¢ r O<kj iplr O<k. I

+

(1 - a

1:

l1-i) Vi:+l(.k

+

~s - (Lr - t)~r) i¢ r 0<"';

+

a

1:

l1-i e(015) { i¢ r k;=O R(U)

Using

R(r.O

+

8(1 - Lr)

1:

PrjV~+lC.k.

+

~s - ~r

+

~j

.l-

(Lf - t)~r - (Lr - t)~i)

j

(13)

1:

fCO

=

1:

E(k) f(i) . where f is a function of i.

j ¢ r j ¢ r O<kj

(14) R(i.lj)

+

aOi -

L)

1:

PjjV~+lC.k.

+

~s - ~i

+

~j - (Lr - O~r - (Lj - Iki )

~ 0

(the induction hypothesis for m = n)

(15) R(r.I)

+

au -

Lr)

1:

PrjV~+l(.k.

+

~s - ~r

+

~j

.l

-

(Lr - t)~r - (Lr - l)~i)

j

+

(1 - a(1 - Lr)) V~+l(

k

+

~s

.l-

(L_{r -} Ikr + ~r)

~ 0

(the induction hypothesis for m

=

n)

we see that

(16) V~.:tfC

+

~s

.l-

(Lf - lkr ) ~ a;

1:

fJ.i e(k) R(i.!)

•

11

+

01

r.

f.J.i e(k) {

so; -

L;)

r.

PijV~+l(.k

+

~s - ~i

+

~j

.l-

(Lr - 1kr - (Li - t)e i)

i

+

(1 - SCIi - Lj

»)

V~+l(.k

+

~s

,l-

(Lr - lkr

+

~i) }

+

(1 - 01

r.

f.J.j €(ki ) V~+l( k

+

~s

.l-

eLf - 1kr)

;;oOr

Applying (16) to the left hand side of (12) results in

+

r.

Prs 01

r.

f.J.i e(ki ) R(d)

s i;oOr

+

r.

Prs 01

r.

J.Li e(kj ) {

s i;oOr

sCt; -

Lj )

r.

PijV~+l( k

+

~s - ~i

+

~j

.l-

eLr - 1)~r - (Li - l)~i) j

+

C1 -

SCIi - L)) V~+l( k

+

~s

.l -

(Lr - t)~r

+

5!.i ) }

If we change the summation over all s and over all j;z!: r and if we add

o

= 01 J.Lr

r.

Prs V~+l(.k

+

~s

,l-

(1'r - 1)~r) S we see that R(dr) +

r.

Prs v~tf(.k + ~s

,l

-

(L_{r -} 1kr ) ~ R(d_r) s i;oO r i;oO r

SOi - Li)

r.

Pij

r.

Prs V~+l(

+

~s - ~i

+

~j

.l-

(Lr - t)~r - (Li - 1)~i ) j S

- (L -_r 1)e _ 1 _ 1

+

e . ) }

+

C1 -

01

r.

f.J.i e(kj ) - 01 p.r)

r.

Prs V~\l(.k

+

~

.l-

(Lr - 1)~r )

i;oOr s

Q' ILrR(r.lr)

+

Q'

r.

ILl e(kj ) R(r.!r)

+

(1 - Q'

r.

ILi e(k_{l ) -} Q ILr) R(r.lr)

I

results in

R(rJr)

+

r.

Prs

vlti(

k

+

.!ts

.l-

(Lf -

Ikr )

~ Q'

r.

ILl e(kj ) R(i.!)

+

01 ILr R(r.lr)

Using s j¢r

+

01

r.

ILl e(k) { s j¢r 8 (lj - ~)

E

Pij ( j R(dr)

+

E

Pes

V~\l(

k

+

.!ts - .!tl

+

.!tj

.l-

(L

r -

O~r

-

(Li - l).!tj») s

+

(1 -

8Cl

i - L) ( R(r.lr)

+

E

Prs Vl+1(

k

+

.!ts

.l-

_{(Lr -}

Ikr

+

.!tj») } s

(17) R(dr)

+

1:.

Prs Vi\l(

k

+

.!ts - ~i

+

.!tj

.l-

_{(Lr -} Ikr - eLi - lki) s

(the induction hypothesis for m ::;: n)

(the induction hypothesis for m ::;: n)

(the induction hypothesis for m ::;: n)

-

13-that is. (12) holds.

Robertazzi. T.G. and Lazar (1985).

On the modeling and optimal flow control on the Jacksonian network. Performance Evaluation 5, 29-43.

Suri, R. (1985),

A concept of monotonicity and its characterization for closed queueing networks, Operations Research 33, 606-624.

Wal, J. van der (1985).

Monotonicity of the throughput of a closed exponential queueing network in the number of jobs, Eindhoven University of Technology, Department of Mathematics and Computing Science, Memorandum COSOR 85-21.

Yao, D.O. (1985),

Some properties of the throughput function of closed networks of queues, Operations Research Letters 3, 313-317.