G-networks with multiple classes of signals and positive customers

EUROPEAN JOURNAL OF ;;W&W$‘kAL

ELSEVIER

European Journal of Operational Research 108 (1998) 293-305

G-networks with multiple classes of signals and positive customers Erol Gelenbe ay*,Ali Labed b~l a Department of Electrical and Computer Engineering, Comp-Sci. and Psychology-Exp., b Ecole des Hautes Etudes en Informatique,

Universitk Renk Descartes,

Duke University, Durham, NC27708-0291. 45 rue des Saints-P&es, 75007 Paris, France

USA

Received 15 March 1996; revised 1 August 1997

Abstract G-networks are queueing models in which the types of customers one usually deals with in queues are enriched in several ways. In Gnetworks, positive customers are those that are ordinarily found in queueing systems; they queue up and wait for service, obtain service and then leave or go to some other queue. Negative customers have the specific function of destroying ordinary or positive customers. Finally triggers simply move an ordinary customer from one queue to the other. The term “signal” is used to cover negative customers and triggers. G-networks contain these three type of entities with certain restrictions; positive customers can move from one queue to another, and they can change into negative customers or into triggers when they leave a queue. On the other hand, signals (i.e. negative customers and triggers) do not queue up for service and simply disappear after having joined a queue and having destroyed or moved a negative customer. This paper considers this class of networks with multiple classes of positive customers and of signals. We show that with appropriate assumptions on service times, service disciplines, and triggering or destruction rules on the part of signals, these networks have a product form solution, extending earlier results. 0 1998 Elsevier Science B.V.

1. Introduction

In this paper we discuss a relatively new class of queueing networks, originally inspired by our work on neural networks, in which customers are either “signals” or positive customers. Positive customers enter a queue and receive service as ordinary queueing network customers; they constitute queue length. A signal may be a “negative customer”, or it may be a “trigger”. Signals do not receive service, and disappear after having visited a queue. If the signal is a trigger, then it actually transfers a customer from the queue it arrives to, to some other queue according to a probabilistic rule. On the other hand, a negative customer simply depletes the length of the queue to which it arrives if the queue is non-empty. One can also consider that

Corresponding author. E-mail: [email protected]. ’E-mail: [email protected].

??

0377-2217/98/$19.000 1998 Elsevier Science B.V. All rights reserved. PIISO377-2217(97)00371-8

294

E. Gelenbe. A. Labed I European Journal of Operational

Research 108 (1998) 293-305

a negative customer is a special kind of trigger which simply sends a customer to the “outside world” rather than transferring it to another queue. Positive customers leave a queue to enter another queue, and they can become signals or remain positive customers. Additional primitive operations for these networks have also been introduced in [ 11.The computation of numerical solutions to the non-linear traffic equations of some of these models have been discussed in [2]. Applications to networking problems are reported in [3]. A model of doubly redundant systems using G-networks, where work is scheduled on two different processors and then cancelled at one of the processors if the work is successfully completed at the other, is presented in [4]. The extension of the original model with positive and signals [5] to multiple classes was proposed and obtained in [6-91. Some early neural network applications of G-networks are summarized in a survey article [lo]. From the neural network approach, the model in [6] was applied to texture generation in colour images in an early paper [ill. The present paper is a generalization of the results presented in [12], where multiple classes of positive customers and signals are discussed. Here we also include multiple classes of triggers. Thus in this paper we discuss G-networks with multiple classes of positive customers and one or more classes of signals. Three types of service centers with their corresponding service disciplines are examined. ?? Type 1: first-in-first-out (FIFO). ?? Type 2: processor sharing (PS). ?? Type 4: last-in-first-out with preemptive resume priority (LIFO/PR). With reference to the usual terminology related to the BCMP theorem [ 131, we exclude from the present discussion the Type 3 service centers with an infinite number of servers since they will not be covered by our results. Furthermore, in this paper we deal only with exponentially distributed service times. In Section 2 we will prove that these multiple class G-networks, with Types 1, 2 and 4 service centers, have product form.

2. The model We consider networks with an arbitrary number n of queues, an arbitrary number of positive customer classes K, and an arbitrary number of signal classes S. External arrival streams to the network are independent Poisson processes for positive customers of some class k and signals of some class c. We denote by /ti,k the external arrival rate of positive customers of class k to queue i and by iii,, be the external arrival rate of signals of class m to queue i. Only positive customers are served, and after service they may change class, service center and nature (positive to signal), or depart from the system. The movement of customers between queues, classes and nature (positive to signal) is represented by a Markov chain. At its arrival in a non-empty queue, a signal selects a positive customer as its “target” in the queue in accordance with the service discipline at this station. If the queue is empty, then the signal simply disappears. Once the target is selected, the signal tries to trigger the movement of the selected customer. A negative customer, of some class m, succeeds in triggering the movement of the selected positive customer of some class k, at service center i with probability Ki,m,k. With probability (1 - Ki,m,k) it does not succeed. A signal disappears as soon as it tries to trigger the movement of its targeted customer. Recall that a signal is either exogenous, or is obtained by the transformation of a positive customer as it leaves a queue. A positive customer of class k which leaves queue i (after finishing service) goes to queue j as a positive customer of class I with probability P’[i,j][k, I], or as a signal of class m with probability P-[i,j][k, rn]. It may also depart from the network with probability d[i, k]. Obviously we have for all i, k:

E. Gelenbe, A. Labed I European Journal of Operational

pjyP+[i,j][k, I] + y+[i,j][k,m] +&$I

= 1.

Research 108 (1998) 293-305

295

(1)

j=l m=l

j=l /=I

We assume that all service centers have exponential service time distributions. In the three types of service centers, each class of positive customers may have a distinct service rate rik. When the service center is of Type 1 (FIFO) we place the following constraint on the service rate and the movement triggering rate due to incoming signals: S

rik +

Ck,m,kk,m = Ci,

(2)

m=l

Note that this constraint, together with the constraint (3) given below, have the effect of producing a single positive customer class equivalent for service centers with FIFO discipline. The following constraints on the movement triggering probability are assumed to exist. Note that because services are exponentially distributed, positive customers of a given class are indistinguishable for movement triggering because of the Markovian property of service time. The following constraint must hold for all stations i of Type 1 and classes of signals m such that Cl=i Cr=, P- b, i][l, rn] > 0 for all classes of positive customers a and 6: &,,>a = f6,rn.b.

(3)

This constraint implies that a signal of some class m arriving from the network does not “distinguish” between the positive customer classes it will try to trigger the movement, and that it will treat them all in the same manner. For a Type 2 server, the probability that any one positive customer of the queue selected by the arriving signal is l/c if c is the total number of customers in the queue. For Type 1 service centers, one may consider the following conditions which are simpler than Eqs. (2) and (3): ria =

rib,

K,m,o = Ki,m,b.

(4)

for all classes of positive customers a and b, and all classes of signals m. Note, however, that these new conditions are more restrictive, though they do imply that Eqs. (2) and (3) hold.

2.1. State representation We denote the state at time t of the queueing network by a vector x(t) = (xi(t), . . . ,xn(t)). Here xi(t) represents the state of service center i. The vector x = (xi, . . . ,x,) will denote a particular value of the state and [xi/ will be the total number of customers in queue i for state n. For Type 1 and Type 4 servers, the instantaneous value of the state xi of queue i is represented by the vector of elements whose length is the number of customers in the queue and whose jth element xij is the class index of the jth customer in the queue. Furthermore, the customers are ordered according to the service order (FIFO or LIFO); it is always the customer at the head of the list which is in service. We denote by ci,t , the class number of the customer in service and by ci,a,, the class number of the last customer in the queue. For a PS (Type 2) service station, the instantaneous value of the state xi is represented by the vector (xi, k) which is the number of customers of class k in queue i.

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Research IO8 (1998) 293-305

3. Main theorem

Let P(x) denote the stationary probability distribution of the state of the network. It is given by the following product form. Theorem 1. Consider a G-network with the restrictions system of non-linear equations: Ai,k

+

described in the previous sections. If the

A;&

qi’ =kri,k + c”,=,Ki,m,k[&,m + I$]

A& =

and properties

TFP-li, il[l,

(5)

’

mlrj,lqj,,,

j=l I=1

has a solution such that for each pair i, k: 0 < qi,& andfor probability distribution of the network state is:

J’(x)=

each station i: cb,

qi,& < 1, then the stationary

Gfigi(*,),

(8)

i=l

where each gi(xi) depends on the type of service center i. The gi(xi) FIFO. If the service center is of Type 1, then

in Eq. (5) have the following form:

PS. If the service center is of Type 2, then gi(Xi)

=I

Xi

1

!fJF.

(10)

1, .

LIFOIPR. Zf the service center is of Type 4, then lbil gi(Xi)

=

(11)

I-Iqi,vi,..

n=l

and G is the normalization

constant.

Notice that Alk may be written as:

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Research 108 (1998) 293-305

291

The conditions requiring that qi, k > 0 and their sum over all classes at each center be less than 1, simply insure the existence of the normalizing constant G in Eq. (8) and the stability of the network.

4. Proof of the main result The proof follows the same line as that for a similar but more restrictive result in [12] which does not cover the case of triggers. The reader who is not interested in the technical details may prefer to skip this section. We begin with some technical Lemmas. Lemma 1. The followingjlow

equation is satisfied:

Proof. Consider Eq. (12), then sum it for all the stations and all the classes, and exchange the order of summations in the right-hand side of the equation:

Similarly, using Eq. (7)

Furthermore,

According to the definition of the routing matrix P (Eq. (l)), we have AkAlk i=l k=l

+ TeA& i=l m=l

=

~~‘1,6Ij-,,(l j=l /=I

Thus the proof of the lemma is complete.

-

d_i,

I]) + ~~~~~4,.re~, i=l k=l j=l

i][l,k]Kj.m,,[~j.m

+ Aim].

/=lm=l

0

In order to carry out the algebraic manipulations of the stationary Chapman-Kolmogorov (global balance) equations, we introduce some notation and develop intermediate results: ?? The state dependent service rates for customers at service center j will be denoted by Mj,l(Xj) where Xj refers to the state of the service center and I is the class of the customer concerned. From the definition of the service rate rj,l, we obtain for the three types of stations.

E. Gelenbe, A. Labed I European Journal of Operational Research 108 (1998) 293-305

298

FIFO and LIFOE’R A4j,/(xi) = PS

rj,l

1iCj,,=I).

Mj,/(Xj) = rjJ(Xj,!/I Xj 1).

the movement triggering rate of class 1 positive customers due to external arrivals of all the classes of signals: FIFO and LIFO/PR Nj,,(xj) = 1 C.,= Et=, Kj,,,,ilj,,. PSN. />I ( xJ .)=( /I .1)-p= k 2 m 1 Jd IF’ XJ,l xJ ?? Aj,/(xj) is the condition which establishes that it is possible to reach state xj by an arrival of the positive customer of class I FIFO Aj,l(xj) = l{c,,w=r), ??

Nj,r(xj) is

LIFWI'R ps

Aj,l(xj) =

Aj,l(xj) =

l{c,,,=[}.

l{~xj.,~>O}~

??

Zj,l,m(Xj)is the probability

??

FIFO and LIFOlF’R Zj,/,m(xi) = 1 {c,,1=I) (Kj,m,l). pS zj,I,m(Xj) = (xj,//I xj I)Kj,m,/. Yj,m(xj) is the probability that a signal of class m which enters a non-empty

that a signal of class m, arriving from the network, will trigger the movement of a positive customer of class 1.

queue, will not trigger the

movement of a positive customer. FIFO and LIFOlPR I;,m(xj) Cf=,

1lc,,, =I}(1- Kj,m,/). ps Yi,m(xj) CFzl(l-Kj,m,~)(xj,~l Ixj I).

Denote by (xj + ej,,) the state of station j obtained by adding to the jth queue a positive customer of class 1. Let (xi - ei,k) be the state obtained by removing from the end of the list of customers in queue, a class k customer if it is there; otherwise (xi - ei,k) is not. defined. Lemma 2. For any Type I, 2, or 4 service center, the following relations hold:

Mj,!(Xj + ej,/)

““~,c:f” =

rj,[qj,l,

J

(13)

J

(14) z~,!,~(x~

+

ej,~)gj’~,(‘ =~” Kj,m,lqj,l.

(15)

J xJ

The proof is purely algebraic. Remark. As a consequence,

0

we have from Eqs. (12), (7) and (13):

(16) and

(17)

E. Gelenbe. A. Labed I European Journal of Operational

Lemma 3. Let i be any Type I, 2, or 4 station, and let Ai

‘;Cxi)=

~4&Z.,(xi) -

Then forthe

+Ni,k(Xj)) +

k(M,,k(X,) k=l

i?i=l

three

Research 108 (1998) 293-305

299

be

cAi,k(Xi)(Aj,k +~4~j)~‘(~~(x,~‘~). I

k=l

types of service centers, l~~X,j,o~di(~,)= C”,=, ,i,rml~,X,,>oI.

The proof of Lemma 3 is in Section 4.1 in order to make the text somewhat easier to follow.

0

Let us now turn to the proof of Theorem 1. The global balance equation of the networks which are considered is:

+

~~~f’(x j=l I=1

i=l

+

j=l

+ ej,/)Nj./(xj

j=l

;=I j=l

ej,l)ob,

I] +

k=l

+ ej.l)Mj,/(xj

+

ej.,)db, I]

k=l

+

ej,l)P(X

-

+

ej,i)f+

-

ej,r)P'b,i][l,

k]A,,k(Xi)l{lx,~,o~

[=l m=l

n/r,,, xJ +

k=l

ei.k +

/=l

+f5?~~s-;~~c ( r=l j=i

?~P(x j=l /=I

I=lm=l

???fI!Li(xj

i=l

+

I=1 m=lh=l

ej,/)P(x

+

ei,k f

ej./ -

eh.s)P-li,

i][l, m]

s=l

We divide both sides by P(x), assume that there is a product form solution, and apply Lemma 2:

E. Gelenbe. A. Labed I European Journal of Operational

300 n

n

R

S

i=l

i=l

j=l

Research 108 (1998) 293-305

j=l

/=lm=l

kc1 I=1

We now apply Eq. (7) to the fourth, fifth, eighth and ninth terms of the second member of the equation: R

pxCALi +4,1(xj)1{lxj\>O} j=l

+Nj,/(xj)l{lx,l>O})

=~~g”~(~j~”

/ii,rAj,l(xj)l{lxjl>O}

j=l

I=1

+

F:FlJ+, Yi,m l (Xi)

i=l

+

{lx,l>O) + ~~qAx.l=o~ i=l m=l

m=l

~,~~$~rj,,qj,,P’ i][l, k]h,k(xi) ~,gi(z,i,;i*) l{lql>o) i=l

j=l

1

k=l I=1

+ i-;~~~,~n,,nKj,n,,qj,,Q~, i=l j=l k=l I=1 m=l

+

I=1

~,~~~1,,Ki,,,kqi,kD[i, i=l

1

i][r,k]~~,k(Xi)gi’~i(x.;i.*’ I

lIl-d>OI

k]

k=lm=l

We group the first, sixth, seventh and ninth terms of the right side of the equation, and pass the last two terms of the first member to the second.

E. Gelenbe. A. Labed I European Journal of Operational

?k(,J) j=l I=1

=

-

+

pt(“i,k(Xi) i=lk=l

Research 108 (1998) 293-305

301

+ N,k(Xi))l(lx,1>0)

L--gi’;i;.y)

Ai,k(Xi)1{lx,I>O}(4k

I

+

n:k)

+

~~~~,,~.rn(Xi)li,*,,>O) i=lm=l

We now apply Lemma 3 to the sum of the first three terms of the second equation.

+~~f~;l,,mK,,m,*qj3*DLil +~~~.*4,.,dli,

)zJ$Aj,/ _ =

~~,~l_l+l>~~

11

i=lm=l

j=l l=lm=l

+ ~~~~ml~~x,i=~~ i=lm=l

11

j=] I=1

+ i-;~~~~~mmK,.rn.k4,.xDlir

kI.

j=lk=lm=l

Now we group the first and fourth terms, and the second and fifth terms of the right side of the equation.

Substituting the values of Db, 11 and

dlj,

I],

and substituting for qj,/ in the second and third terms and grouping them we have:

eFAj,l

=

j=l I=1

i-;Fl,+

pk*j,*

i=lm=l

j=l /=I

+

ekA;*- eFA,t,- ~~,l,~m j=l I=1

j=l I=1

which yields the following equality which is obviously satisfied, ~~~j,l

j=l

I=1

=

k?AjJ:

j=l I=1

concluding the proof.

i=lm=l

302

E. Gelenbe, A. Labed I European Journal of Operational

Research 108 (1998) 293-305

As in the BCMP [13] theorem, we can also compute the steady state distribution of the number of customers of each class in each queue. Let yi be the vector whose elements are (_vi,ki,k) the number of customers of class k in station i. Let y be the vector of vectors bi). We omit the proof of the following result. Theorem 2. Zf the system of Eqs. (S)-(7)

n(~)=

has a solution then, the steady state distribution n(y) is given by

fib(i),

(18)

i=l

where the marginal probabilities hibi) have the following form:

(19)

4.1. Proof of Lemma 3

The proof of Lemma 3 consists of algebraic manipulations of the terms in the balance equations related to each of the three types of stations. LIFOZPR. First consider an arbitrary LIFO station and recall the definition of di:

=1 (/xa~>O)~Ai,k(Xi)(~i,k k=l

l{lxi>O}di(Xi)

-

l(lx.I>ojkNi.k(Xi)

+ /ick)

+

gi(xi

VdUes

-

l{~xI~>O]~~i,k(Xi)

k=l

1~,~,,>0~~~,,6,rn(Xi).

k=l

We substitute the

- ei’k) gi(Xi)

m=l

of I&, Mi,k,Ni:,kand Ai,k for a LIFO station: R

l{lx,~>O}4(Xi)

R

=1 Ikl>o}~ll=,.~=k)(ni,k k=l R -

Use the

S +

1+~>0~~&~1

m=l

l{,x,,>O)~&l m=l

as

l(,,i>o)C1(~i,,=k~ri,k k=l

k=l

Ic~.I _-kl

(1 -K,m,k).

??I=1

value Of qi.k from Eq. (5) and some cancellations of terms to obtain.

=

and

-

S

l{ix~l>o}~l~r,,~=kJ~~i,m.k~i,m

k=l We

+ /lck)/qi,k

l{l.,+o)

l{lx,l>O}di(xi)

~~=‘=,

=

k= 1 ‘=“’-k’

l{,,,=k} = l{~x,l>~l,we finally get the result l(~x/>Oj~i.,.

m=l

E. Gelenbe, A. Labed I European Journal of Operational Research 108 (1998) 293-30.5

303

FIFO. Consider now an arbitrary FIFO station: l{lx,l>O}di(Xi)

=

1 {~x,~>O)~Ai.k(Xi)(A,k k=l

+

ATk)gi(xi

- ei’k) Si(xi)

l(~.x~>O]~M,.k(Xi) k=l

s

R

-

-

c l{lx’I>olNi’ +‘{l~,l>o)~~,yi;,(Xi). k(Xi) Wt=l

k=,

Similarly, we substitute the values of &,,, M;,k,Ni,k,Al,k and qi,k:

’ {~.rd>O}di(xi)

=

Yi,k f

l(~x,~>Oj~l{c,,x=k) k=l

-

‘~,x,l>O~~l~,,,=k)‘,*k=l

f

‘~ls,>O)~l,,~l~,,,,=a)(l In=1

kKi,tn,k&m WFl

+

eKi.m,kli, m=l

‘jx,,>o)~lj~,,,=ki~K,,,,kr:im k=l

-

m=l

-Ki.m,k). k=l

We separate the last term into two parts, and regroup terms,

R -

+

ri,k

1{~x,l>O]~1{c8,,=kj k=l

l{,xt,>O&E.,,&{c, m=l

+

eKi,m,k 1i.m +

t?l=l

2Ki,m,kATm m=l

,=k). k=l

Conditions (2) and (3) imply that the following relation must hold.

Thus, as ~{I~,I>o}

cfEl l{c,,l=k}

=

~{Ix,I>o)~

we finally get the expected result.

(21) m=l

PS. Consider now an arbitrary PS station.

l(lx,j>O)di(xi)

=

1 (,s,>O)~Ai,k(Xi)(Ai,k k=l

-

$l{~x.f>O)N,k(Xi)

AS usual, we substitute the

VdUeS

+

A:k)gi(xi

gi(Xi)

+

l(ji.,>O~~ii.,~,-(Xi). m=l

of &,, kfi,k, Ni,k,Ai,k.

ei’k)

-

l{[x~>O)~~i,k(Xi) k=l

304

E. Gelenbe. A. Labed I European Journal of Operational

+

Research 108 (1998) 293-305

l{,xi,>o)f+-r %(l - Ki,m,k). ‘? Ixil m=l

k=l

Then, we apply Eq. (5) to substitute qi,k. After some cancellations of terms we obtain. R l{lx,l>O}4(%)

=

1 (..,>O~~~~~i,~,kn,

+

1 {l~~l~Ol~~~&~~~

c1 -

6m,k).

m-l

Finally we have

(22) Since

l{lx;l>O}

cf==,(Xi,k/bil)

proof of Lemma 3.

=

l{lxil>O}y

once again, we establish the relation we need. This concludes the

Cl

PI W. Henderson, Queueing networks with negative customers and negative queue lengths, Journal of Applied Probability 30 (3) (1993).

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[18] E. Gelenbe, G-networks with triggered customer movement, Journal of Applied Probability 30 (3) (1993) 742-748. [19] E. Gelenbe, G-networks with signals and batch removal, Probability in the Engineering and Informational Sciences 7 (1993) 335342. [20] W. Henderson, B.S. Northcote, P.G. Taylor, Geometric equilibrium distributions for queues with interactive batch departures, Annals of Operations Research 48 (l-4) (1994). [21] E. Gelenbe, A. Labed, G-networks with multiple classes of customers and triggers(to appear).