An Approximate Analytical Method for General Queueing Networks

transcript

IEEE TRANSACTIONS ON SOFTWARE ENGINEERING, VOL. SE-5, NO. 5, SEPTEMBER 1979

An Approximate Analytical Method for GeneralQueueing Networks

RAYMOND A. MARIE

Abstract-In this paper, we present an approximate solution for theasymptotic behavior of relatively general queueing networks. In theparticular case of networks with general service time distributions (i.e.,fixed routing matrix, one or many servers per station, FIFO discipline),the application of the method gives relatively accurate results in a veryshort time. The approximate stationary state probabilities are identi-fied with the solution of a nonlinear system. The proposed method isapplicable to a larger class of queueing networks (dependent routingmatrix, stations with fimite capacity, etc.). In this case, the structureof the network studied must satisfy certain decomposability conditions.

Index Terms-Approximate solution, equilibrium distribution, per-formance evaluation, queueing networks, queueing theory, service timedistribution.

I. INTRODUCTION

FOR some years now, a considerable effort had been de-rvoted to the problem of obtaining analytic solutions for

more and more general queueing networks. However, exactanalytic solutions for complex networks are still limited tothose which satisfy local balance equations, i.e., networks inwhich the transition probabilities are fixed and in which thestations (with unlimited waiting room)

1) have exponential service time distributions, or2) contain more servers than customers, or3) possess a queueing discipline such that the behavior of

the station with nonexponential service time distribution isthe same as the behavior it would have with an exponentialservice time distribution having the same mean.These restrictions justify the search for approximate meth-

ods which has been undertaken over the past few years. Dif-fusion techniques [11, [21 and iterative techniques [31, [4]allow networks in which the transition probabilities are fixedand in which each station is composed of a unique server witha general service time distribution to be treated. These meth-ods obtain the marginal asymptotic probabilities of the statesof each station, and from these, the basic system performancemeasures may be obtained.To model more general networks, simulation or numerical

methods are usually employed. Each of these methods hasserious disadvantages. Simulation techniques allow complexnetworks to be studied, but computation time limitationsimply that for large networks, the results will not be accurate.

Manuscript received March 8, 1978; revised February 27, 1979.The author is with the Institut de Recherche en Informatique et

Systemes Aleatoires, I.N.S.A., 35031 Rennes Cedex, France.

Numerical methods (e.g., [5]) give accurate results, but arerather limited with respect to the size of the network.The method proposed in this paper obtains an approximate

analytical solution to relatively general queueing networks.Maybe the main interest of the paper is that the approximatesolution is defined as being the solution of a complex systemof nonlinear equations, rather than as the results of some moreor less empirical algorithm using an auxiliary network. Thisfact enables us to prove that the approximate solution satisfiessome basic properties such as the invariance of the total num-ber of customers and the Chang-Lavenberg Theorem on workrates. Given the system of equations, the final solution isobtained through an iterative process.

In order to facilitate understanding, the method is first pre-sented (Section II) for the particular case of a network in whicheach station has a unique server with a general service time dis-tribution (the discipline being FIFO). This method uses theunderlying concept of the conditional throughput of a queuewith general service time distribution.The following section (Section III) considers the general-

ization of the method. The properties which the networksmust possess before the proposed method is applicable areconsidered.

Finally, the last section is devoted to conclusions and tosome numerical examples which demonstrate the accuracyof the method.

II. STUDY OF THE NETWORK WITHGENERAL SERVICE TIMES

Let R be a closed network containing m service facilities andN customers for which the following holds.

1) The routing matrix T = (Pij) is fixed, i.e., the probabilitypi] that a customer leaving station i enters station i is indepen-dent of the state of the system.2) Each station contains only one server (rj = 1). The ser-

vice times at station i, i= I, , m are distributed with mean0 <1-' <+0 and general distribution function Fi(t) havinga rational Laplace transform.3) Within each station, the queueing discipline is first come,

first served.Let e = (ni,n2, -- ,fnm) be a state of R where ni is the

number of customers in station i.By definition, we denote by X(n)ICkIrIN anMIGIr/FIFOIN

queue in which the arrival process X(n) depends on n, thenumber of customers in the queue. Ck denotes a service time

MARIE: ANALYTICAL METHOD FOR QUEUEING NETWORKS

distribution with a rational Laplace transform having k pcThe notation is chosen with respect to the Coxian represeition and in accordance with the Ek notation.

Basic HypothesisMathematically, the proposed method is based on the

lowing hypothesis.Hypothesis 1: An approximate solution to the asympt

probabilities of the states of network R is given by the piabilities defined by the following system of equations:

x,K1(N - i) Q,(i - 1)pi(i) =K(N-i+ 1)X Q,(i)

i=l,- ,N

if n, = 0

if n1 >0a) Ai(n;) = ni

les.nta-

Fig. 1.

ULtU ymptotic behavior of networkR to be close to the behavior ofrob- network R*. Here, the properties of the conditional through-

put of a X(i)/CkIl queue are used.In order to give some explanations, let us consider first the

well-known Coxian representation (cf. [6] ) of the service law(1) of type Ck as shown in Fig. 1.

Let {Nt, Jtj denote the Markovian process associated withthe queue X(n)/CkIIIN and which takes its values in { [0, 1,

(2) AN] X [1,2,- ,k]}.Let (n,j) denote a state of that process. (Nt = n if there are

n customers in the queue, and Jt = i if the server is in fictitiousphase j.)Let q(n, j) be the asymptotic probability of the state (n,j).By considering the Chapmann-Kolmogorov linear equations

(3) of the process {Nt, Jt} in asymptotic behavior, it is easy toshow (cf. [41 ) that we have for such a X(n)/Ck/l IN queue

b) x = (xl,x2, ***,xm) is a solution ofx.=x

c) K(N)= EA

and L2 ni=Ni=i

d) K. (u) = En, A(j) (5)

and E ni = ui=l

and n. = 0

e) Q1(-) is the asymptotic probability of the states ofthe queue [X(i)/Ck1I1N] j; this queue has the same

service time distribution as station i ofR, and where

x) K (N- i-i)K,(N- i)

Some Important RemarksP(e) is also the asymptotic probability of the state of a

closed exponential network R * containing N customers andin which each station j has an exponential server with state-dependent service rate v,(n1), n, = 0, * * *,N for j = 1, * - *, m.Thus, all the characteristic values obtained from P(e) may becalculated as for an exponential network by means of classicalalgorithms (assuming that the functions A(-) are known).Physically, Hypothesis 1 implies that we consider the real as-

where, in accordance with the notations of Fig. 1,

k£ Mjbjq(i, )

v(i) 1=1k

E q(i,j)j=l

Note that this relationship is introduced in system (I) bymeans of the relationship (2).

Properties of the Solution ofSystem (I)In this section we will derive some properties of the system

(I) which will be used in order to obtain the solution.Let Pj(i) be the approximate marginal probability obtained

by means of system (I), i.e.,

P(i0) P(e)n, * nm

mand E n. =N

and nt = i.

Taking the product form ofP(e) into account, we have

xA K-(N- i)

I =1-a =1-a =1-aI

1 1 2 2 3 3 k

v(i) QO) = X(i 1) QY 1)

P,(i- 1) Aj(i) XK(N-i+1)

Pj(i) A(iY-1)'NxjKj(N -i) @

That is, by using (3) and (6),

P,(i- 1) v,(i)Pj(i) )Mji - l )

Consequently, by considering the relationships (2), (6), and(8), we obtain

Pj(i- 1) Qj(i- 1)

Pj(i) Qj(i)which involves the equality between probabilities:

P,(i) = Q(i) i=0, *,N. (10)By once again taking the form of P(e) into considera

we have

£ £ iPj(i)=N.j=1 i=0

This relationship, along with relationship (10), implies ti

£ iQ,(i)=N.1=1 i=l

This last relationship establishes that the sum of the r

values of customers in queues [X(i)fCk/l /N] j, j = 1,equal to N.

If Pi(-) denotes the exact marginal asymptotic probabilistationj, the work rate of stationj may be written as

U1= I -Pj(O).

For the type of network R which is considered heretheorem of Chang and Lavenberg (cf. [7]) shows thawork rates of the stations are related by

We will now show that the probabilities given by system (I)are such that

(1 - Pj(0)) x- Ili(I - P,(O) xigi

i.e., that the calculated work rates (which are also those ofR *)are in the same ratio as the exact work rates of networkR.Since P*(-) is also the marginal asympf )tic probability in

networkR *, we have

- NXi = E Pi(k)Pi(k).

Moreover, since R * is an exponential network, we also have

X; ~Xj (13)

On the other hand, it follows from (2), (6), and (10), that

Xik= kPi(k)Pi(k)= 1 vi (k) kk=l k=1

£ Xi(k)Q,(k).k=O

Thus, Xi is also the mean flow through station [X(i)/Ck/ I/N] j.Then

Xi=pj[1 - Q1(O)] =/.ll - P,(O)1 j= 1, *,m (14)which yields the final results by taking relation (13) intoaccount.

Solution ofSystem (I)System (I) is solved by the following iterative method:

(11) J0 =°,0 if i>0

hat l wdI)(i= I-IX) () X >(Q 1( )

(12)and for

i = 0, ... ,Nj= 1, ,m.

The iterative process is halted when relationships (12) and(14) are almost satisfied, i.e., when

m NN- Z iQ1(i)

j=1 i=0 <e (16)N

immi/n E i

ImLIri= 1,2,o ,m (17)

1 - QO(0)ri= ~-l jf= I, *. MXi IPi

For the experiments so far performed with this model, thisinitialization has permitted the solution to be obtained in onlya few iterations.

III. GENERALIZATIONLet R be an ergodic closed queueing network which contains

N customers, and in which the transition probabilities may bestate dependent. R is composed of m queues, some of whichmay have limited waiting room.

Fig. 2.

I C~b 1b

Fig. 3.

na la nb Pb

Fig. 4.

Before stating the solution for such a network, it is neces-sary to first introduce a decomposition of the network intosubnetworks, and secondly to calculate the flow betweensubnetworks.LetI= {1, , m} be the set of indices of the stations ofR.The partition {J(k)}1 6 k < s Of I defines a decomposition of

R in s subnetworks R1,** ,Rk,* R,. The vector e = (n1,I** nm) represents the states of R and the vector ek those

ofRk.Let us decompose R in a maximal number S of subnetworks

in such a way that1) state-dependent transition probabilities pij(.) must only

depend on the state ek of the subnetwork Rk containing sta-tion j2) if a customer enters a queue ofR k by means of a state-

dependent transition probability, it is necessary that thiscustomer, on leaving Rk, enters any station j, VI E I\J(k) witha probability which is independent of the path followed in thesubsystem3) if the station j of Rk has a limited waiting room, cus-

tomers entering station j must come from the stations of Rkonly.

Figs. 2, 3, and 4 give examples of such subnetworks.These conditions imply that the transition probabilities irij

among subnetworks must be independent of the states of the

subnetworks. If there exists an S such that S > 1, it is there-fore possible to determine an equation of conservation offlows between subnetworks.Let a = (a,, -- , as) be the normalized vector representing

the flow through the subnetworks Rl,. * * ,Rs. This vectoris obtained as follows.

1) IfR contains only fixed transition probabilities and if xis a solution ofx T= x, we have immediately

2) If R contains dependent transition probabilities (cf. Figs.1 and 2), it is sufficient to take for each subnetwork Rk thetransition probabilities pi = pii(ek) of an arbitrary state ekand to calculate the unique normalized solution x' of x' "=x'; using conditions 1) and 2) we have

atk jEPi(ki E- J(k) i E- I\J(k) /

If S > 1, the proposed method may be expressed, as in Sec-tion II, by means of a hypothesis on the behavior ofR.Hypothesis 2: Let e = (N1, * ,* NS) be a state of R where

Nk is the number of customers in subnetworkR k*An approximate solution for the asymptotic probabilities

of the states of the networkR is given by the probabilities de-

1--N I

Ctk= F, xi 2: Pii .

i EE J(k) i EE I\J(k)

fined by the following system of equations:

j1 (A$(Nj))K(N)

a,Kj(N- i) Q1(i- 1)v1(i) K,(N - i + 1) Q1(i)

if Nj = 0

a),Y(i) if Nj>O

b) a= (al, - - * , cis) is calculated byscribed above

c) K(N))=

Vi,VjE7Z and II>1:

Prob{t(t + dt) = i +jIt(t) = i} = O(dt)

where 0 (dt) denotes a quantity for which lmdtm0 (O(dt)/dt) =(19) 0. t(t) is the stochastic process associated with the states E1

of the partition {EJ}o0 i< N.In this particular problem, Ei is the set of the states 7r of R

(20) such that there are i customers in the system.By letting Q(i) = asymptotic probability that the state 7r of

d belongs to Ei[v(i) dt + 0(dt)1 = Prob {t(t + dt) = i - I It(t) = i}

(21) and by recalling that

the method de-

Is(iAj(

N1,* ,NS\IlAj(Nj)/(22)

sand Ni =N

N,,K.(u

,lISZ lAy (23)NI -,NS\ i=' Aj(NTp

sand E Ni = u

andN, = 0

e) Q1(Q) is the asymptotic probability of the state of anopen network R9 having identical structure to RI(and the same parameters) but having a Poissonarrival process with state-dependent rate:

xiK-(N - i- 1)K1(N - i) (24)

Here, according to the nature of R,, the probabilities Qj(j)may be calculated either by an exact (or approximate) analyti-cal method, by a recursive technique using the Markoviangraph associated with R,? (by means of fictitious stages in thecase of general service laws), or finally by a numerical method.

Remark

Once again, relationship (20) of system (II) allows us to usea property of the conditional throughput of any subnetworkRJ, j=1,* ,S. In order to give a physical meaning toHypothesis 2, we will now state this property.Let R be the irreducible and ergodic Markovian system as-

sociated withR. Let E be the set of states q of R.For the system 6R considered here, there exists a partition

{Ei}o 6 i6N ofE such that

[) (i) dt + 0(dt)] = Prob {t,(t + dt) = i + 1 1 t(t) = i}we have the following theorem.Theorem: For the Markovian system 6:

X(i) Q(i) = (i+ 1) Q(i + 1). (25)

The proof of this theorem (cf. [8] ) is obtained by establish-ing first that in asymptotic equilibrium, we have

X(i)P(i) + v(i)P(i) = X(i - l)P(i - 1) + v(i + 1)P(i + 1).

Then, taking the limit condition into account, X(0) P(0)=)(1) * P(1), we get the relation (25).This relationship means that, for any partition {Ei}o i<N

which satisfies the above property, the process t(t) convergesasymptotically in law to a birth/death process.

Properties of the Solution ofSystem (II)As in Section II, if we defined the semimarginal probability

PjQi)= P(e)Nl, -,Ns

Sand I'=N

andNj = i

we have

(26)Pj(- 1) pi(i)P(i) X,(i-1)

and, by using (20) and (24), we obtain

Pj(i-) Q1(i- 1)P,(i) Q,(i)

Pi(0) = Q(i) i = O, INwhich implies, as in Section II, that

S NjiQ )=N.

...i= 1, 'Ni = s

Similarly, by using relationships (20), (24), and (28), it maybe shown that

_ N K(N-1)k=o

X(k) P(k) K(N)

and thus

X1 a,_I Q,= 1,.. 'S*,. (30)

Solution ofSystem (II)For the solution of this system, an iterative method similar

to that of Section II is used. According to the nature of R,the initialization, i.e., the determination of the sequence{Pilo)(-)}, j = 1, *,S can be more or less complex.

°)( ) is calculated by means of a simplified subnetworkRJ constructed from R, according to the following rules.

1) The service distributions of type K are replaced by ex-ponential service distributions (having the same mean).2) Limited waiting rooms are replaced by unlimited waiting

rooms.3) State-dependent transition probabilities pi,(j) are re-

placed by the fixed transition probabilities pi' introducedabove.The sequence {X(o)(i)} is then quickly obtained by means

of the classical algorithms of exponential networks.The iterations are then performed by means ofthe relationship

aKiK(11)(N- i) 1)X--1

K=0,*-- ,Nand for (31)

j=1, ,S.

Using (29) andfollows:

(30), the test-termination conditions are as

Case ofan Open NetworkIn the case of an open network for which the arrival rate

Xo(n) of the Markovian arrival process may depend on thetotal number of customers n in the network, an approximatesolution may be obtained by considering the canonicallyassociated closed network having N customers, N being suf-ficiently large to ensure that the probability of the sourcecontaining zero customers is negligible.

IV. TEST RESULTS AND CONCLUSIONS

Although no mathematical proof of convergence has beengiven, numerous experiments with different models andwith different parameters failed to find an example whichdid not converge.Tests were conducted with a view to determining the speed

of convergence and the accuracy of the method.

Speed ofConvergenceTwo elements must be examined: the execution time for a

single iteration and the number of iterations.If the networkR is such that S = m and r1 = 1, the execution

time of an iteration is very short because of the following.1) As mentioned above, the sequences {Xj(n1)} are com-

puted by an algorithm similar to the one used for the normal-izing constant of a closed exponential network (having stationsfor which the service rates are state dependent).2) Computation of the marginal probabilities is effected by

means of efficient recursive methods.In the case where numerical methods are used to determine

the stationary state probabilities of subnetworks R5, it is thecomputation of these semimarginal probabilities which isdominant in the execution time of a single iteration.The number of iterations depends on m,N, the type of sub-

networks Rk, and the value given to e. Generally, e is chosenequal to 10-3 or 10-4. According to the type of modeltested, the method converged in most cases in three to fiveiterations. However, in some very unfavorable cases, the num-ber of iterations becomes higher (e.g., 18 iterations). Moreparticularly, it is the case of networks containing many sta-tions in series.

AccuracyFor the purpose of obtaining a measure of the accuracy of

the method, the models used were chosen small so that exactnumerical solutions could be obtained (cf. [5] ).The proposed method can only be of interest if it produces

better results than those obtained by using simpler models,(33) e.g., an exponential model. In this respect, all the tests were

convincing: the results of the method are clearly much closerto the exact results than the results of the exponential net-work canonically associated with R. This is demonstrated inthe examples in the following. The first example (cf. Fig. 5)deals with a central server model. The central unit is modelizedby a hyperexponential server having a high variance service dis-tribution. The servers of the I/O units are Erlang-2. Compared

ri- E ri

I si= 1, **,S

ri = ji= 1,.*.*,S.xj

STATION 1 2 3

PARAME TE

TYPE OF LAW F2 E2 E2

MEAN _1 2 1

COEFF.K2 100 0.5 0.5I 1

NUMBER OF CUSTOMERS: N = 6VALUE OF £ *1os 4

NUMBER OF ITERATIONS; 7

x-- -X--

RESULTS RELATIVE ERR. IN %

STATION 1 2 3 1 2 3

(a) 0.6719 0.6718 0.3359 -0.21 -0.21 -0.21

Ui (b) 0.6732 0.6732 0.3366 - -

(C) 0.8349 0.8349 0.8349 24 24 24

(a) 2.5600 2.9223 0.5175 0.91 0.56 -7.09

n i (b) 2.5371 2.9060 0.5569 - -

(C) 2.6606 2.6606 0.6788 4.87 -8.44 21.88

Ui : work-rate of station i

ni : mean queue length of station i

(a): solution obtained by the proposed method

(b): exact solution

(C): "exponential solution"

"exponential" results

exact results

results of the proposed method

Fig. S. Central server model.

to the exponential modelization, the results of the proposedmethod are relatively near the exact results, more particularlywhen the marginal probabilities are considered. It can benoted that, for this experience, the value of e had been taken

to 10-4 and that the iterative process had stopped after seveniterations. By taldng e= 10-6, we would get approximatelythe same results, but after 11 iterations.

In the second example (cf. Fig. 6), it must be noted that,

TAT IONS1 2 3

PARAMETERS

TYPE OF LAIW H2 H2 E2_

MEAN 1 1 1

COEFF. K2 10 10 0.5

NUMBER OF CUSTOtIERS : N

VALUE O' c : 10

NUMABER OF ITERATIONS : 5

U; : work-rate of station i

ni; mean queue length of station i

(a) solution obtained by the proposed method

(b) exact result

(C) : "exponential solution"

"exponential" results

exact results

results of the proposed method

STATIONARY PROBABILITY DISTRIBUTION OF STATION N°3

0.4 0.3 0. 2 ~~~~~0.1 0

STATIONARY PROBABILITY DISTRIBUTION OF STATION Nol (and 2)

N=4P3 (n3)

0.1~~~~~N

0.4 0.3 0.2 0.1 o

Fig. 6. Symmetrical structured network.

if the service laws were exponential, the third marginal dis- work and formulation for the problem of building an approxi-tributions of probabilities would be identical. mate solution for a large class of queueing networks (in partic-

ular, when the marginal probability distributions are needed).V. CONCLUSION However, further studies would be necessary in order to in-

We have presented an approximate iterative technique which vestigate the mathematical aspect of the accuracy of the ap-produces relatively accurate results and which provides a frame- proximation and of the convergence of the iterative process.

The algorithm has been implemented for networks withfixed routing matrix; the implementation for state-dependentrouting is currently taking place.

ACKNOWLEDGMENTThe author would like to express his thanks to the anony-

mous referees for their constructive criticisms.

REFERENCES

[1] H. Kobayashi, "Application of the diffusion approximation toqueueing networks, Part I: Equilibrium queue distributions,"J. Ass. Comput. Mach., vol. 21, pp. 316-328, Apr. 1974.

[2] E. Gelenbe, "On approximate computer system models," J. Ass.Comput. Mach., vol. 22, pp. 261-269, Apr. 1975.

[3] K. M. Chandy, U. Herzog, and L. Woo, "Approximate analysisof general queueing networks," IBM J. Res. Develop., vol. 19,pp. 43-49, Jan. 1975.

[4] R. A. Marie, "Methodes iteratives de r6solution de modeles math-6matiques de systemes informatiques," RAIRO Inform./Comput.Sci., vol. B2, no. 2, pp. 107-122, 1978.

[5] W. J. Stewart, "MARCA: Markov chain analyser," Institut deRecherche en Informatique et Systemes Al6atoires de Rennes,France, Res. Rep. 45, June 1976.

[6] D. R. Cox, "A use of complex probabilities in the theory ofstochastic processes," in Proc. Cambridge Phil. Soc., vol. 51,1955, pp. 313-319.

[7] A. Chang and S. S. Lavenberg, "Work-rates in closed queueingnetworks with general independent servers," Oper. Res., vol. 22,pp. 838-847, 1974.

[8] R. A. Marie and W. J. Stewart, "A hybrid iterative-numericalmethod for the solution of a general queueing network," inProc.3rd Symp. on Measuring, Modelling and Evaluating Comput. Syst.,Bonn, West Germany, H. Beilner and E. Gelenbe, Eds. Amster-dam: North-Holland, Oct. 1977, pp. 173-199.

Raymond A. Marie received the Doctoratd'Ingenieur and the Doctorat d'Etat es-sciencesmath6matiques from the University of Rennes,Rennes, France, in 1973 and 1978,respectively.He is currently working at the Probability

Laboratory of the Institut de Recherche enInformatique et Systemes Aleatoires, Rennes.His research interests are in the areas of queue-ing theory with application to the performanceof computer systems, and in the modeling oflarge maintenance organizations.

Correspondence

Correction to "An Experiment in Software Error DataCollection and Analysis"

N. F. SCHNEIDEWIND ANDHEINZ-MICHAEL HOFFMANN

In the above paper' there was an inadvertent error in theheadings of Tables V and VI. The table in the upper right-hand corner of page 281 should be labeled Table V and shouldread: Software Error Experiment. Correlation Coefficients(Error Properties Versus Complexity Measures).

Manuscript received June 18, 1979.N. F. Schneidewind is with the Naval Postgraduate School, Monterey,

CA 93940.H.-M. Hoffmann is with the Federal German Navy, Philipp-Lassen-

Koppel 38, 2390 Flensburg, Germany.1N. F. Schneidewind and H.-M. Hoffman, IEEE Trans. Software

Eng., vol. SE-5, pp. 276-286, May 1979.

U.S. Government work not protected by U.S. copyright.

An Approximate Analytical Method for General Queueing Networks

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