AN ANTI-BLOCKING CONTROL POLICY FOR TANDEM QUEUEING … · Blocking phenomena may appear in any queueing network with limited capacity queues. We propose a simple admission control

AN ANTI-BLOCKING CONTROL POLICY FOR TANDEMQUEUEING NETWORKS

Jean-Claude Hennet 1and Khaled Smaili 2

1 LAAS-CNRS. 7 Av. Colonel Roche. 31077 Toulouse. FRANCE. E-mail: hennet@laasfr

2 University of Lebanon. BP 135 292. Beyrouth LEBANON

AbstractBlocking phenomena may appear in any queueing network with limited capacity queues. We propose asimple admission control policy, to decrease the risks of blocking which deteriorate the systemperformance. Under classical Markov assumptions, the controlled system is exactly modelled in the caseof two tandem queues, and approximately modelled for more than two queues. The quality of theapproximate analytical model is then assessed by comparison with simulation results. It is establishedthat in most cases, the performance of the controlled system is much higher than that of the uncontrolledsystem.

1 Introduction

Blocking phenomena are very common in practice in production systems, computersystems and communication systems. They are generally associated with tandemsubnetworks in which the saturation of a queue blocks its upstream server. ln aproduction line with large size items, for instance, a buffer may often happen to be full.And, in the absence of a control, the blocking may rapidly propagate to severalmachines, causing a global decrease of the performance and availability of the system[Dubois, Forestier 1982]. Admission control devices are thus needed to reduce theinfluence of the limits on buffer sizes. ln general, networks with finite queues do notsatisfy the BCMP [Baskett et al.1975] assumptions, and their steady-state probabilitydistribution does not admit a product-form. However, under markovian assumptions onarrivaIs and service times, the optimal admission policy can be ca1culated for simpletandem networks. It has been shown that it is of the switch-over type [Rosberg et al ..1982], but the switching surface does not have any generic property. So, it has to becalculated in each particular case.

Even when they are not controlled, systems with blocking are generally difficultto analytically evaluate, and approximate methods have been specifically elaborated forthis purpose [Gershwin 1987; Dallery, Frein 1993].

The main objectives of this paper are to reduce blocking by a simple statedependent control and to propose an analytical technique to evaluate the performance ofthe scheme. A control device is supposed to be located at the input of the consideredtandem subnetwork. It is able to admit customers or to redirect them to othersubnetworks. The proposed controllaw is particularly simple and general. It consists ofrejecting (redirecting) customers arriving whenever at least one of the queues is full, andto admit them in any other case.

A remarkable result is that, under markovian assumptions, the controlled systemadmits a product-form for the basic case of 2 tandem servers. Furthermore, this basiccase can be approximately replaced by a single server. This reduction can then begeneralized to a tandem network with m servers in cascade. The approximate

Proc. INRIA/IEEE Symposium on Emerging Technologies and Factory Automation (ETFA’95), Paris (France), 1995, pp.549-556.

performance of the controlled system is then validated and compared to the evaluatedperformance of the uncontrolled system, for sorne cost values attached to the states andto the rejection action.

2 The model

The considered model is represented on Fig.1. It consists of m stations in cascade.Each station is made of a buffer (queue) and of a service center (server). The capacityof the first queue is supposed infinite. The following stations i (2 :::;i :::;m) have finitecapacities, Ni. Customer arrivaIs are supposed to occur according to a Poissonprobability law with mean rate 1.Each customer is served in sequence at aIl the stations,and leaves the system after service at the mth service center. The priority discipline ateach server is supposed to be FIPO (first in first out). The service time distribution for

each server k (1 :::;k :::;m) is supposed exponential with mean service rate ~k' The

analysis can be easily extended to state-dependent mean service rate ~k(nk)' The state ofthe model is described by (nI (t), ... , nk(t), ... , nm(t),), where nk(t) is the number ofcustomer in station k at time t. A control device is supposed ta be located at the inputstream, before the first station. Its role is ta enable or to disable the entrance to thetandem queue. An arriving customer may then either go to the first queue or directlyleave the system (and possibly go to an other route). The studied control policyconsists of refusing admission if and only if at least one of the stations is full (n 1(t)=N ifor sorne i (2 :::;i :::;m». Any admitted customer must proceed along the tandem networkuntil completion of the last service.

Redirection to other

sub-systems

If::3 ilsuch that ~=N i~

Figure 1 : A controlled tandem network with m servers

3 The case of two tandem queues

Consider first the case m = 2. An arriving customer is accepted and goes to queue 1 ifn2 < N2. Otherwise, if n2 = N2. it is rejected. The state of this system at time t is (nl(t),n2(t». Its Markov chain is represented on Fig. 2.

Let i be the number of customers on station 1, andj the number of customers onstation 2. The system state is determined by (i, j). The balance equations associatedwith the steady-state probabilities of the system states are as follows :

Àp(O, 0) = ~2P(0, 1)

[À + ~l]pCi, 0) = ÀpCi-1, 0) + ~2P(i, 1) for i ~ 1

[À + ~2]P(0, j) = ~lP(1,j-1) + ~2P(0, j+1) for 1:::;j < N2 X'1)

~2P(i,N2)=~lPCi+1,N2-1) for i~O

[À+~l +~2]P(i, j) = ~lPCi+ 1, j-1) + ~2PCi,j+ 1) + ÀpCi-1, j); i ~ 1 and 1:::;j< N2


112

Figure 2 : Markov chain of the 2 servers tandem network

It is not difficult to check that the set of equations (1) is consistent with thefollowing factorization : ..

pO, j) = C ( ~l Y ( ~ J (2)

The invariant state probabilities have a product-form, in terms of the parameters

of the servers, III and 112' Parameter A in expression (2) is the mean arrivaI rate, and Cis a normalization constant which can be calculated from equation (3) :

00 N2

L L pO, j) = 1. ( 3)i=Oj=O

For Pi = (Nil), ( i = 1,2), constant Chas the following expression:

( 1 - P2 )C = (1- Pl) N +1 . (4)

1 - P22

Remark : Without a control, the invariant probabilities do not have a product form.The product form (2) is due to the control action. So, in sorne sense, thecontrol decouples the two stations.

Under the product-form, the controlled system steady-state performance can becomputed from the performance of each station alone. The first queue is M/M/1, andthe second one is M/M/1/Nz [Kleinrock 1975].The mean number of customers in thesystem is given by :


(5)

To compare the performance of the two stations tandem system with andwithout control, let us select an average cost function with a blocking cost, cb, applyingto both cases, and a rejection cos t,cr, attached to the admission control.

The rejection probability is equal to the probability of an arrivaI when thesecond queue is full. At the equilibrium, under the property of independence of PoissonarrivaIs from the state of the queues, the rejection probability per time unit, denoted byPf(2)., takes the value:

(6)

ln the considered case of after service blocking, the blocking probability isequal to the probability of an end of service at the first server when the intermediatebuffer is full. The blocking probability per time unit at the equilibrium is :

(7)

Existence of a unique invariant probability measure shows the ergodicity of thecontrolled Markov chain. Under the selected cost function, the time average is equal tothe space average [Yosida 1980]. The infinite horizon average cost, denoted by J, takesthe form :

J = ÀcrL p(i, N2) + 11lCbL p(i, N2)i=O ~1

Using expressions (5) and (6), the cost function J can be written :

(8)

For particular values of the system parameters and of the cost parameters, theoptimal value of the average cost criterion under all possible policies, J*, and theoptimal admission/rejection policy, can be computed by a c1assical technique, such aspolicy iteration. On the other hand, the equilibrium probabilities of the uncontrolledsystem can be approximately computed by solving the set of equilibrium balanceequations for a large (but not infinite) value of NI. Then the cost function for theuncontrolled system, Jo, can be evaluated. Experiments have shown that, for nonnegligible values of Cb, the cost function J under the proposed anti-blocking policy, isgenerally much doser to J* than to Jo [Hennet, Smaili 1993].

4. An approximate model of the 2 stations tandem system

ln order ta generalize the study to an m stations tandem system (m>2), the 2 stationstandem system will now be approximated by a single station model. The approximatemodel will be obtained by allowing the mean arrivaI rate and the mean service rate tobe state dependent.

To obtain unbiased mean system performance, the probability of state i of theaggregate system will be defined as the sum of the probabilities of states (i-k, k) of the


original 2 stations tandem system. Using such astate aggregation, the mean arrivaIrates and service rates for the approximate system are obtained from the state graph ofthe original system (Fig.2).

Let Àag(i) denote the mean arrivaI rate of the aggregate model in the presence of

i customers. And llagU) is the mean service rate of the aggregate model in the presence

of j customers. Let Pag(i) be the probability of having the aggregated system in state i.Using relations (2) and (9), the following properties can be easily derived:

V n~ 0,(9 )

(10)

pG,O) J~P(j--k,k)[ PCi-N2,N2)] [

ÀagCN2) = Â 1 Nz and llagU) = 112 1

L pCi-k,k)k=O

The state probability transition graph of the aggregated model is represented on Fig.3.

Figure 3 : Markov Chain of the aggregated model

The steady-state probabilities of this chain are as follows :

(2)00

Combined with L Pages) = 1 , it gives :5=0

(13)


5. Evaluation of an m stations tandem system.

Consider now the m stations tandem system of Fig.1. This system can beevaluated by successively analyzing the (m-1) sub- systems obtained by the aggregationtechnique of section 4. Namely, stations 1 et 2 are first aggregated into the

approximate model FI, using expressions (11) to compute Àa (.) and j.lag(')' Moregenerally, at stage i (2::; i ::;m-1), model Fi is obtained from mo~el Fi-l and station i+ 1.The number of customers in models Fi and Fi-l, respectively denoted by Siand Sj-b arerelated by: Si= Si-l+ ni+l (Fig.4).

Redirectian ta ather

sub-systems

Àag(Si) Fi j.lag(S)~

Figure 4. : Construction of aggregated model Fi

Note that the product form of state probabilities is obtained only at the firstaggregation stage. For subsequent stages, (2 ::;i ::;m-1), the mean arrivaI rate at stationi+ 1 depends of the state of Fi-l' State probabilities are then approximated for a largebut limited size of the queue, Nag(i-1), for model Fi-l'

Analysis of model Fm-l gives approximate performance of the line:

- the mean throughput, Àa,which is equal to the mean~rate for model Fm-l:N (m-l)

ag

À = ~foll À (sî (14)a L...i -el ag m-Ys =0

m-l

- the mean number of customers, En

- the percentage, Po of customers re-directed to other subnetworks

N.g(m-2) ( Nm )~= ~ L [i + j]Pm-1 Ci, j) .

1=0 J=O

where Pm-l(i, j) is the steady-state probability of having Sm-2=i, and nm=j.

(15)

Other performance indices can be evaluated from the approximate models takenbackwards. ln particular, for 2::; k ::;rn-l, the probability for queue k to be full, denotedby Pr(k+l)"is given by :

N.g(k-I)

Pr(k+I) = [1- Pr(k+2)] L Pk Ci, Nk+l); with Pr(m+I) = O.i=O

(16)

Finally, when reaching the first stage backwards, the product-fonn can be usedto compute Pr(2) and the performance indices of servers 1 and 2 for the 2 stationstandem model. An approximate value of the cost function, J, for the anti-blockingpolicy is thus given by :


J

m-l

ÀCrPr + L ~kCb(kl r(k+1)k=l

(17)

(18)

N-700

with Pr(k+1)= [1-Pr(k+2)] LPkn, Nk+1).i= 1

Cb(k)is the elementary blocking cost for server k. ln the absence of any admissioncontrol, the total cost is :

m-l

J 0= L ~kcb(k)Pb(k)k=l

N2 Nk Nk+2 Nm

with Pb(k) =L L'"L L'" LP(nl,n2, ... ,nk,Nk+1, ... ,nm).n) =0 n2=0 nk=l nk+2=0 nm=O

ln practice, the blocking probability for server k in the uncontrolled case, denoted byPb(k) can be computed by simulation, or by an analytical technique such as the oneproposed by [Dallery, Frein 1993].

6 Comparison of the costs with and without control

Consider the example of a production line with 4 machines in cascade (m = 4) andfinite capacity intermediate buffers. The data is as follows : mean arrivaI rate for parts l

CÀ = 1), mean service mi for machine i ~ = 1.6 for i = 1,2,3,4), capacity Nk for stationk (queue + server) (Nk = N for k = 2, 3, 4), unit rejection cost Cr (cr = 2) and blockingcost CbU)for machine j (Cb(j)= 2 for j = 1,2, 3).

Curves CI and C3 on figure 6 represent the variations of costs J and Jo asfunctions of the common size of the buffers, N. Curve C2 uses the same data and thesame control policy as curve CI, but the cost evaluation is obtained from simulation.The c10seness of these two curves, compared to curve C3, validates the choice of theanalytical technique, much faster than simulation, for evaluating the interest of theproposed admission policy.

0.4

035

0.3

0.25

0.2

0.15

0.1

0.05

CI COST WITHCONTROL(ANALYTICAL)C2 COST WITH CONTROL (SIMULATION)C3 COST WITHOUTCONTROL (SIMULATION)


References

F. Baskett, KM. Chandy, R.R. Muntz, F.G.Palacios, 1975 "Open, Closed, and MixedN etworks of Queues With Different Classes of Customers", Journal of theAssociation for Computing Machinery; vol 22, N° 2, pp. 248-260.

Y. Dallery, Y. Frein, 1993, "On decomposition methods for tandem queueing networkswith blocking" Operations Research, vol. 41, 2, pp. 386-399.

D. Dubois, J.P. Forestier, 1982, "Productivité et en-cours moyens d'un ensemble dedeux machines séparées par une zône de stockage" RAIRO Automatique, 16,2, pp.105-132.

S.B. Gershwin, 1987, "An efficient de composition methodfor the approximateevaluation of tandem queues withfinite storage space and blocking" Oper.Res. 35,2, pp.291-305.

J.c. Hennet, K Smaili, 1993, "Decentralized control of admission into productionpaths" E.C.C. 93, volume 1, pp. 164-169.

L. Kleinrock, 1975, "Queueing systems", Volume 1: Theory, Wiley, New York.

Z. Rosberg, P.P. Varaiya, 1. Warland, 1982, "Optimal control of service in tandemqueues" IEEE Trans. Autom. Control, vol. 27, No 3, pp. 600-610.

K Yosida, 1980, "Punctional Analysis ", Sixth Edition, Springer Verlag.


AN ANTI-BLOCKING CONTROL POLICY FOR TANDEM QUEUEING … · Blocking phenomena may appear in any queueing network with limited capacity queues. We propose a simple admission control

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