Matrix geometric analysis of the shortest queue problem with threshold jockeying

Matrix geometric analysis of the shortest queue problem with

threshold jockeying

Citation for published version (APA):

Adan, I. J. B. F., Wessels, J., & Zijm, W. H. M. (1991). Matrix geometric analysis of the shortest queue problem with threshold jockeying. (Memorandum COSOR; Vol. 9124). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1991

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Memorandum COSOR 91-24 Matrix-geometric analysis of the shortest

queue problem with threshold jockeying I.J.B.F. Adan

J. Wessels W.H.M. Zijm

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Eindhoven, October 1991 The Netherlands

MATRIX-GEOMETRIC ANALYSIS OF THE SHORTEST QUEUE PROBLEM WITH THRESHOLD JOCKEYING

IJ.BF. Adanl), J. Wessels1).2), W.HM. Zijm3)

Abstract: In this paper we study a system consisting ofcparallel servers with possibly dif-ferent service rates. Jobs arrive according toa Poisson stream and generate an exponen-tially distributed workload. An arriving job joins the shortest queue, where in case of multi-ple shortest queues, one of these queues is selected accordingtosome arbitrarY probability distribution.Ifthe maximal difference between the lengths of thec queues exceeds some threshold valueT, then one job switches from the longesttothe shortest queue, where in case of multiple longest queues, the queue loosing a job is selected accordingtosome

arbi-traryprobability distribution. It is shown that the matrix-geometric approach is very well suitedtofind the equilibrium probabilities of the queue lengths. The interesting point is that for one partitioning of the state space an explicit ergodicity condition canbederived from Neuts' mean drift condition, whereas for another partitioning the associated

R

-matrix can bedetermined explicitly. Moreover, both partitionings used are different from the one sug-gested by the conventional way of applying the matrix-geometric approach. Therefore, the paper can beseen as a plea for giving more attention to the question of the selection of a partitioning in the matrix-geometric approach.

Key Words: jockeying, matrix-geometric solution, queues in parallel, shortest queue.

1. Introduction

The matrix-geometric approach, as introduced by Neuts in his book [8], has proved to be a

powerful tool for the analysis of markov processes with large and complicated state spaces,

par-ticularly the ones that appear when modeling queueing or maintenance systems. One stage in

the approach is a partitioning of the state space. Usually this stage does not get much attention.

Users tend to use so-called natural partitionings which are suggested by the way of modeling or,

if such a natural partioning is not available, they select a partitioning that fits most elegantly

with the boundary behaviour of the process.

In

a previous paper

[1],

the authors demonstrate

already that in the case of the shortest queue problem with threshold jockeying for 2 parallel

queues it is more effective to apply a partitioning based on the behaviour of

the

process in the

1) Eindhoven University of Technology, Department of Mathematics and Computer Science, p.o. Box 513,5600 MB - Eindhoven, The Netherlands.

2) International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria.

3) University of Twente, Department of Mechanical Engineering, p.o. Box 217, 7500 AE - Enschede, The Netherlands.

interior of the state space than based on the boundary behaviour as has been proposed by Gertsbakh [4].

The present paper investigates the shortest queue problem with threshold jockeying forc

parallel queues. The results in this case are even more striking, since, against the belief that the matrix-geometric approach provides any insight in the case c

>

2 (cf. Zhao and Grassmann [10]), we show that, by using the right criterion for selecting a partitioning, two partitionings can be constructed that help in solving the equilibrium equations completely. Actually, this paper has two goals. In the first place it presents a simple way of showing; that the shortest queue problem with threshold jockeying for

c

parallel queues has a nice and elegant solution.In the second place it demonstrates that the use of a proper criterion for selecting a partitioning of the state space can be crucial for the success of the matrix-geometric approach.

Consider a queueing system consisting ofcparallel servers. The service rate of serveriis

'Yj,

i

=

1, ...,

c,

where, for simplicity of notation, it is supposed that

'Yl

+ ... +'Yc

=

c.

Jobs arrive according to a Poisson stream with rate cp and generate an exponentially distributed worldoad with unit mean. An arriving job joins the shortest queue. Incase of multiple shortest queues, one of these queues is selected according to some arbitrary probability distribution. Ifthe maxi-mal difference between the lengths of the c queues exceeds some threshold value T, then one job switches from the longest to the shortest queue. In case of multiple longest queues, the queue loosing a job is selected according to some arbitrary probability distribution. This system can be represented by a continuous time Markov process whose state space S consists of the vectors(n1, n2, ..., nc ) wherenj is the length of queuei, i= 1, ...,c. Due to the threshold

jock-eying the state space S is restricted to those vectors for which

I

nj - nj

I

~T for alliand j. Forc

=

2 this model has been analyzed by Gertsbakh [4] and by Adan, Wessels and Zijm [1]. For arbitrary cZhao and Grassmann [10] use the concept of modified lumpability of con-tinuous time Markov chains to find the equilibrium distribution of the queue lengths. The spe-cial case ofinstantaneousjockeying(T

=

1) has been analyzed by Haight [6] for c

=

2 and by Disney and Mitchell [2], Elsayed and Bastani [3], Kao and Lin [7] and Grassmann and Zhao [5] for arbitrary c. .

In this paper it is shown that the matrix-geometric approach developed by Neuts [8] is very well suited to analyse this problem.Insection 2 we show for a suitably chosen partitioning of the state space that an explicit ergodicity condition, which is obviously

P

<

1, can be derived from Neuts' mean drift condition. However, for that partitioning the associated R-matrix cannot be determined explicitly. Therefore, in section 3 we propose another partitioning for which the associated R-matrix can easily be determined explicitly. Actually, the latter choice generalizes the choice used in [1] for c

=

2, which was suggested by a more direct way of solving the equili-brium equations. Gertsbakh [4] uses the matrix-geometric approach forc =2, but his choice for the partitioning does not lead to an explicit solution for the associated R-matrix.

-3-2. Necessary and sufficient ergodicity condition

Application of the matrix-geometric approach requires a partitioning of the state space. First define for 1=0, 1, sublevel Ias the set of states (nit ..., nc)eSsatisfying _nl+...+nc

=

I. Then for each state(nlo , n_{c )}at sublevelI

>

(e-I)Tnone of the queues is empty and the tran-sition rates from this state are identical to the rates from the corresponding state (nl+l, ..., nc+l) at sublevelI+c. This suggests to define for all I

=

0, 1, ... level Ias the union of the sublevels Ie, le+l, ..., le+c-l and to partition the state space S according to these levels withI =T, T+1, ... and to put the levels 0, 1, ...,T-1 with less regularbeha~iourinto one set The states at level I are ordered by sublevel, states from each sublevel being ordered lexico-graphically. For this partitioning the generatorQis of the following fonn, where the first class corresponds to the group oflevels 0, ...,T-1,

Boo BOl 0 0 0

BIO Al A o 0 0

Q= 0 _{A 2} Al A o 0

0 0 A 2 Al A o

Corresponding to the partitioning of level I~T into the sublevels Ie, le+l, ..., le+c-l, the square matrices A0,A1and A2 are of the fonn

Ao,o AO•I 0 0

A I•o AI,I A I•2 0

o

A2,I A2.2 A2,3

o

(1)

The Markov processQis irreducible and, since two states at levels

>

Tcan reach each other via paths not passing through levelsST, the generator A o+A1+A2is also irreducible. Thus theorem 1.7.1 in Neuts' book [8] canre~dilybe applied. Specifically the Markov processQis ergodic if and only if

where e is the column vector of ones and 1t is the solution of

1t(A o +A l +A 2)=O, lee=l.

By partitioning 1t as (1to, ..., 1tc-l) and the column vector e as

e=[~O],

1;;-1

(3)

corresponding to the form(1)ofAo, A1andA2'we get as a result that inequality(2)reducesto

and the equations(3)to

1ti-lAi-l

. .

j

+

1tjA i i

+

1ti+lAi+l

.

j =0, i=1, ..0' c-2 ,

1tc-2Ac-2.c-l

+

1tc-lAc-l.c-l

+

1toAO.c-l =0 .

(4)

(5)

Since the flow from a state at sublevel1

>

(c-1)Tto sublevel 1+1 is cp (an arrival) and the flow

tosublevel1-1 is c (a service completion) it follows that

AO.lel =cpeo, Ao.oeo=-e(p

+

l)eo, AO.c-lec-l = ceo ,

Ai,i+lej+l =Cpei, Ai,iej= -e(p

+

l)ei' Ai,i-l ei-l= cei' i= 1, "0'c-2 ,

Ac-l,oeo = cpec-l' Ac-l,c-l ec-l = -e(p

+

l)ec-l' Ac-l,c-2ec-2= cec-l 0

Hence inequality (4) simplifiesto CP1tc-l ec-l

<

c1toeo

and multiplying the set of equations(5)with ei leads to CP1tc-l ec-' - p(c

+

1)1toeo

+

C1tl el =0 ,

CP1tj_l ej-l - p(c

+

1)1tjei

+

C1tj+l ej+l =0, i= 1, ...,c-2 ,

(6)

CP1tc-2ec-2 - p(c

+

l)1tc-l ec-l

+

c1toeo =00

By the symmetry of these equations it follows that 1toeo = ... = 1tc-l ec-l and thus from (6) we can conclude that:

Theorem1: The Markov process

Q

is ergodic

if

andonlyifp

<

1.

So, for the chosen partitioning of the state space, the mean drift condition(2)easily leads to the desired ergodicity condition, but the associated R-matrix, however, cannot be determined

-5-explicitly. Inthe next section we adapt the definition of the levels and show for this new parti-tioning that the associated R-matrix can be determined explicitly.

3. Explicit determination of R

We now adapt the definition of level I as the set of states (n1t ••• , nc)eS satisfying

max(nit ...,nc)

=

I. The state space S is partitioned into the sequence oflevels T, T+l, ....The

levels 0, 1, ...,T-1 with less regular behaviour are put together in one set The states at each level are ordered lexicographically. For this partitioning the generatorQis of~eform

Coo COl 0 0 0 CIO Cn Do 0 0

Q=

0 Dz DI Do 0

0 0 _{D z D}1 Do

The square matricesDo,D1andD

_z

are of dimensions

m xm

where

m

is the number of states at a level ~T. The Markov processQ is irreducible and, since two states at levels

>

Tcan reach each other via paths not. passing through levels:$; T, the generator D

_o

+D1+Dz is also

irreduci-ble. Thus theorem 1.7.1 in Neuts' book [8] can again be applied. By partitioning the equilibrium probability vector

P

into the vector

(Po, ..., Pr

-1)and into the sequence of vectors PT,

Pr

+1, ••.

where PI is the equilibrium probability vector of levell, we then obtain

PI=PrRI-r , I> T ,

where the matrix R is the minimal nonnegative solution of the matrix quadratic equation

Do +RD_{1+RzD z}=0. (7)

(8) The matrixR can be determined explicitly due to the special matrix structure of Do. Since it is only possible to jump from level Ito level1+1 via state (I, I, ...,I), it follows thatall rows of

Do are zero, except for the last row. ThusDo is of the form

Do

=

[~

]

where v

=

(vo •...•

Vm-I) .

Since rows in R which correspond to zero rows in Do, are also zero (see e.g. the proof of theorem 1.3.4 in [8]), we conclude that R is also of the form

R

=

[~]

wherew=

(wo ...Wm-l).

Hence the matrix-geometric solution simplifies to

(9)

LetV, be the set of states(nl •...• nc)eSsatisfyingnl+...+n_c=1. Then the componentWm -] is

detennined by balancing the flow betweenV, andV,+_ltyielding forI> (c-l)T

P(V'+l)c =P(V,)CP.

and by applying this relation

c

times.

P(V'+c)

=

pC

P(V,) .

On the other hand. the setVcl is a subset of the union of the levelsI. 1

+

1. ...•1+T-1.so it

fol-lows from (8) that forI

>

T

P(Vcl)=Kw~_] •

for some constantKbeing independent ofI.Combining(9)and(to)yields

(10)

The remaining components of w are solved from equation(7).which. by insertion of the special fonns of R and Do. simplifies to

v+w(DI +Wm_ID 2)=0.

Finally. substitutingWm-I

=

pC

leads to w=-v(D] +pcD2)-I.

where the inverse ofD]

+

pC

D2 exists. since this matrix can be interpreted as a transient gen-erator (escape is possible at least from the last state).

Theorem 2: R=

[~]

where _{w=-v(D t}

+

P'D2)-t.

Remark:

The explicit solution of R is mainly due to the special matrix structure of Do.Infact, Ramaswami and Latouche [9] show that if the generatorQcan be partitioned as in the begin-ning of this section andDo is given byDo

=

x'ywherexis a column vector andyis a row vec-tor. then R is explicitly detennined. once its maximal eigenvalue is known.

4. Conclusion

Inthis paper we studied the shortest queue problem withcservers and threshold jockeying and we showed that the matrix-geometric approach is very well suited to analyse this problem. The interesting point was that a proper choice for the state space partitioning depends on where one is interested in. One partitioning leads to an explicit ergodicity condition and for another partitioning the associated R-matrix is detennined explicitly. Another conclusion may be that. for the application of the matrix-geometric approach for solving the equilibrium equations of markov processes. the aspect of selecting a proper partitioning is of crucial importance. Inthe

-7-treated case the extension of gertsbakh's partitioning to

c

>

2 does not provide any insight. However, two other partitionings, based on the process behaviour in internal points, lead to a complete solution of the problem.

References

1. ADAN, I.J.B.F., WESSELS, J., AND ZIJM, W.H.M., "Analysis of the asymmetric shortest queue problem with threshold jockeying,"Stochastic Models, vol. 7, 1991 (to appear).

2. DISNEY, R.L. AND MITcHELL, W.E., "A solution for queues with instantaneous jockeying and other customer selection rules,"Naval Res. Log., vol. 17, pp. 315-325, 1971.

3. ELSAYED, E.A. AND BASTANI, A., "General solutions of the jockeying problem,"Eur.J.

Oper. Res., vol. 22, pp. 387-396, 1985.

4. GERTSBAKH, I., "The shorter queue problem: A numerical study using the matrix-geometric solution,"Eur.J.Oper. Res., vol. IS, pp. 374-381, 1984.

5. GRASSMANN, W.K. AND ZHAO, Y., "The shortest queue model with jockeying," Naval Res. Log., vol. 37, pp. 773-787,1990.

6. HAIGHT, F.A., "Two queues in parallel,"Biometrica, vol. 45, pp. 401-410, 1958.

7. KAo,E.P.C. AND LIN, C., "A matrix-geometric solution of the jockeying problem,"Eur.

J.Oper. Res., vol. 44, pp. 67-74, 1990.

8. NEUTS, M.P.,Matrix-geometric solutions in stochastic models, Johns Hopkins University

Press, Baltimore, 1981.

9. RAMASWAMI,V. AND LATOUCHE, G., "A general class of Markov processes with explicit matrix-geometric solutions,"OR Spectrum, vol. 8, pp. 209-218, 1986.

10. ZHAO, Y. AND GRASSMANN, W.K., "SolVing a parallel queueing model by using modified lumpability," Research paper, Queen's University, Dep. of Math. and Stat, 1991.

P.O. Box 513

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M.W.I. van Kraaij W.Z. Venema

J. Wessels

M.W.I. van Kraaij W.Z. Venema

J. Wessels

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of the shortest queue problem with threshold