Annals of Operations Research 48( 1994)493-511 493

Geometric equilibrium distributions for queues with interactive batch departures

W. H e n d e r s o n , B.S. N o r t h c o t e and P .G . T a y l o r

Teletraffic Research Centre, Department of Applied Mathematics, University of A delaide, Adelaide, Australia

Gelenbe et al. [1, 2] consider single server Jackson networks of queues which con- tain both positive and negative customers. A negative customer arriving to a non- empty queue causes the number of customers in that queue to decrease by one, and has no effect on an empty queue, whereas a positive customer arriving at a queue will always increase the queue length by one. Gelenbe et al. show that a geometric pro- duct form equilibrium distribution prevails for this network. Applications for these types of networks can be found in systems incorporating resource allocations and in the modelling of decision making algorithms, neural networks and communications protocols.

In this paper we extend the results of [1, 2] by allowing customer arrivals to the network, or the transfer between queues of a single positive customer in the network to trigger the creation of a batch of negative customers at the destination queue. This causes the length of the queue to decrease by the size of the created batch or the size of the queue, whichever is the smallest. The probability of creating a batch of negative customers of a particular size due to the transfer of a positive customer can depend on both the source and destination queue.

We give a criterion for the validity of a geometric product form equilibrium distri- bution for these extended networks. When such a distribution holds it satisfies partial balance equations which are enforced by the boundaries of the state space. Further- more it will be shown that these partial balance equations relate to traffic equations for the throughputs of the individual queues.

Keywords: Product form, geometric distribution, batch departures.

I. Introduction

J a c k s o n n e t w o r k s wi th single server queues have g e o m e t r i c p r o d u c t f o r m

equ i l i b r ium d i s t r ibu t ions [5]. G e l e n b e [1] i n t r o d u c e d the c o n c e p t o f nega t ive cus to -

m e r s to these n e t w o r k s . W h e r e a s a n o r m a l (or pos i t ive) c u s t o m e r increases the

queue length o f a n y queue it jo ins by one, a nega t ive c u s t o m e r a r r iv ing to a n o n -

e m p t y queue will r educe the length o f t ha t q u e u e by one c u s t o m e r , bu t has no effect

when a r r iv ing a t a queue which is a l r eady e m p t y .

© J.C. Baltzer AG, Science Publishers

494 IV. Henderson et al./Geometric equilibrium distributions

Applications for these types of networks can be found in many fields. In resource allocation problems an arrival of a positive customer could represent a request for a set of resources, and an arrival of a negative customer could repre- sent a decision to cancel such a request. Several communications protocols may also be modelled using queueing networks incorporating negative customers. For example consider the "Go back N " protocol for data packet transmission. When transmitting messages between buffers using this protocol, the source buffer stores N data packets until it receives an "all clear" signal indicating that all packets have arrived intact at the destination buffer. To model the number of data packets stored in a network of communicating buffers, positive customer arrivals correspond to the arrival of data packets to a buffer, and negative cus- tomer arrivals correspond to an "all clear" signal deleting N customers from the buffer. In another application, consider the length of a queue to represent the input potential to a neuron. Then positive and negative customers may repre- sent excitatory and inhibitory signals respectively in the modelling of neural net- works [6].

Gelenbe gave conditions for these networks, with Poisson arrival streams and exponential service times, to have geometric product form equilibrium distri- butions. Gelenbe et al. [2] analysed single server queueing systems with positive and negative customers, allowing different service distributions. In doing so, queue discipline had to be taken into account since service times were no longer exponential, and assumptions were made as to which customer is forced to leave a queue by the arrival of a negative customer to that queue. For each of these networks, when a positive customer completes its service and is transferred to another queue, there is a fixed probability that it will become a negative customer.

In this paper, we generalise the Gelenbe [1] network of queues by allowing any arrival of a positive customer to a queue to generate the simultaneous arrival of a batch of negative customers to the queue. The batch size is determined by an arbitrary probability distribution, and includes the zero option in which the queue size is increased by one (a single positive customer arrival). We give a con- dition for this more general network (consisting of N queues) to have a geometric product form equilibrium distribution. When this happens the N parameters of the distribution satisfy a set of 2 N equations which are imposed by the bound- aries of the state space (the hyperplanes {ni = 0, for i E S, S C_ ( 1 , 2 , . . - , N } } , where n,.-- number of customers in queue i). These equations are shown to be equivalent to a set of N traffic equations representing expressions for the throughputs of the N servers in the network. Thus the parameters of the geo- metric distribution can be identified as the throughputs of the queues. Further- more, it is established that the global balance equations for the process can always be written in terms of (N + 1) partial balance equations which are indivi- dually satisfied by the equilibrium distribution of the network.

W. Henderson et al./Geometric equilibrium distributions 495

2. The model

The model consists of a network of N single server queues. Arrivals of posi- tive customers to queue i from outside the network occur in a Poisson stream with rate Ai and request an exponentially distributed service time with parameter #i. Fol- lowing a service completion at queue i, with probability dij(k), for k > 0, the depart- ing customer will be transferred to queue j, simultaneously triggering the arrival of a batch of k negative customers to queue j. This could be thought of as a request for the elimination of k customers from queue j, due to a service completion at queue i. In response to such a request, k customers will be eliminated from queue j, or if there are less than k customers in queue j following the transfer, then queue j will be emp- tied in response to the request. We also allow arrivals of positive customers to queue i from outside the network to trigger the creation of a batch of k negative customers at queue i with probability doi(k), (i.e. consider the outside of the network to be labelled as queue 0). In particular, customers move from queue i to queue j as posi- tive customers with intensity Izidij(O) and positive customers arrive to queue i from outside the network with intensity Aido~(O). Single customers depart the network from queue i leaving the rest of the network unaffected either by a positive arrival to queue i releasing 2 customers from queue i (with rate ~idoi(2)), or by a service completion at queue i being transferred to queue j (for any j ) and immediately released as a negative customer (with r a t e #idij(1)). Although this means that many events have the same net effect on the network we have found that the equa- tions governing the process are more compact with this notation. We assume that either Aidoi(O) > 0 or #jdji(O) > 0 for at least o n e j E {1 ,2 , . . . , N} for each queue i, so that it is possible for the number of customers at queue i to increase by 1. This ensures the irreducibility of the state space.

The parameters dij(k) of the system are probabilities and so

N oo

~ dij(k )=1 , f o r i E { 1 , 2 , - - - , N } , (1) j = l k = 0

oo

doi(k)=l , f o r i E { 1 , 2 , . . . , N } . (2) k = 0

To avoid further duplicity of intensities, and for simplicity of notation, we assume that dii(k) = 0 and dio(k ) = 0, Vi, k, i.e. customers may never be trans- ferred to the queue at which they were last served, nor may they create negative cus- tomers outside the network following a service completion. To model systems for which we require dii (k) ~> 0 consider the following. A customer that is transferred to the queue at which it was served enabling the creation of a batch of k negative customers has the same net effect on the network as an arrival from outside of the network to that queue enabling the creation of a batch of k 4-1 negative

496 W. Henderson et al./Geometric equilibrium distributions

customers. Thus we can model systems with d i i ( k ) > 0 by artificially increasing the arrival rate of positive customers to queue i, and rescaling the d0i (k) probabilities. Set d i i ( -1 ) = 0 for notat ional convenience and introduce new parameters A~ and d~i(k) which satisfy

A*id~i(k + 1) = Aidoi(k + 1) + #idii(k), k >_ - l .

In the new model, let queue i have arrival rate and transfer probabilities A7 and d~i(k ) respectively, given by

O0

A7 = Ai + #i Z dii(k)' k=O

d~i(k) = -~i (Aid°i(k) + #idii(k - 1)), k _> O.

This augmented model will be equivalent to the previous model .

3. The equilibrium distribution

At time t queue i contains ni(t ) customers. We form a con t inuous time Markov chain { n ( t ) = (nl (t), n 2 ( t ) , . - . , n u ( t ) ) : t >_ 0}, which satisfies the usual C h a p m a n - K o l m o g o r o v equations. We look for an invariant measure of the form

N

p(,,) = K I ] (3) i=1

K can be chosen so that this sums to one iff Yi < #i for i = 1 , . . . , N. In this case we show that Y/can be interpreted as the t h roughpu t of the single server at queue i, that is, the por t ion of the arrival stream that actually receives a complete service f rom the server.

Consider the state space d'I = {m : m i ~ 0, Vi E { 1 , 2 , . . . , N }} and for each m E ~ let S _c { 1 , 2 , . . . , N } be the set of queues which are empty in state m, i.e. m i = 0 if i E S, m i > 0 if i E { 1 , 2 , " ", N }\S. To leave state m one of the following transit ions mus t occur:

a positive cus tomer arrives to a non-empty queue f rom outside the net- work, and has a net effect on the number of cus tomers at that queue at rate

Ai(1 -- d0i(1)); i~_s

W. Henderson et at . /Geometric equilibrium distributions 497

a lone positive customer joins an empty queue from outside the network at rate

Z iCS

• a service completion occurs at a non-empty queue at rate

Transitions into state m occur from states

• m - ei due to the arrival of a lone positive customer from outside the network at rate

Aidoi(O);

m + leg due to a positive customer arriving with l + 1 negative customers to a queue from outside the system, not causing the queue to empty, for l > 1 at rate

~_a Aid° i ( l + 1); i~.s

• m + le~ due to a positive customer arriving with l + 1 negative customers to a queue from outside the network, causing the queue to empty, for l _> 1 at rate

Z G iES k = l + l

m + eh + (l -- 1)e; due to a positive customer being transferred from queue h to queue i with I negative customers, not causing queue i to empty, for l _> 0 at rate

mdhi(l); i¢s

m + e h + (l - 1)e i due to a positive customer being transferred from queue h to queue i with I negative customers, causing queue i to empty, for l > 1 at rate

iES k=l

498 W. Henderson et aL/Geometric equilibrium distributions

Thus the global balance equations relating flux into and out of state m are

P( ' ) Ii~S /~i(l - dOi(l)) -Jr Z /~idOi(O) "~ Z i~S ~ ZP('-ei)'~id°i(O)iq[s

+ Z Z p(m+lei)Aid°i(l+ 1)+ Z Z p(m+lei) Z /~id°i(k) i~s /=1 iEs /=1 k = l + l

~_,lZh ZP(m+eh +(l-1)ei)dhi(l) h = l l=0

Z Z p(m+eh+(l-1)ei) din(k) "

iES 1=1 k=l

+

+ (4)

Here, e, is an N x 1 vector of zeros except for a 1 in the ith position. Substituting from eq. (3) and simplifying (dividing by p(m)) gives:

A,(1 - d o i ( 1 ) ) + ~ Aidoi(O ) + ~ Izi = ~ (#i/Yi)Aidoi(O) iq[S iE S iqlS igS

o¢) oo + Z Z ( Y i / # i ) l A i d ° i ( l + l ) + Z Z (Yi/~i)l Z Aid°i(k)

iq[s /=1 iEs 1=1 k = / + l

+ Z Yh Z(Yi/#i)t-'dhi(l) + Z Z (Yi/#i)l-' din(k) • h = l l = 0 iES 1=1 k=t

( 5 )

That is, if the equilibrium distribution is to be given by eq. (3) then {Y,., for i = 1 ,2 , . . . , N } must satisfy eq. (5). We refer to eq. (5) as the equation imposed by the boundary of the state space given by the hyperplane {m:m~ = 0, i E S, m; > 0, i ¢ S}. Note that eq. (5) depends upon the state of the network only through those queues which are empty and in its present form represents 2 N equa- tions, one for each S c_ {1 ,2 , . . . , N}. In the following we show that these 2 N equa- tions can be replaced by N state independent equations, one for each queue in the network. Rearrange eq. (5) so that all of the EiCs terms are on the LHS, and all of the Ei~ s terms are on the RHS, so that we require

where

Z Ai = Z Bi, i~S iES

Ai = Ai(1 - d0i(1)) +#i- (#i/Y,)Aidoi(O) N

- + l ) - rh dh,(/), I=1 h = l I=0

W. Henderson et al./Geometric equilibrium distributions 499

B i = -Aidoi(O) + Z (YffPi)t Z Aid°i(k) /=1 k = l + l

N oc c~

h = l l=1 k=l

Lemmas 3.1, 3.2 and 3.3 establish some basic relationships involving the expressions Ai and Bi. The results of these lemmas are utilised in the proof of theorem 3.1, which is the main result of this paper.

LEMMA 3.1

N oo (x3

(Y,/,,)(A,+ B,) = r , - a,do,(O)- ~ r~d~,(O) + ~ ( r , / . , ) ' h = l l=1 k = l + l

N 00 oo

+ Z ~ Z(~,/~,) ' Z d~,(k), h = l l=1 k = l + l

Aidoi(k)

Proof

(Y,/.,.) (Ai + B;) = (F,/U,)~,(1 - do~(1) - do,(O)) + Y, - ~doi(O)

(x) N o o

-- Z ( r i / . i ) I+l /~idoi(l q - 1)- Z Yh Z ( Yff#,)t dh,(l) l=l h=l t=0 oo oo N oo oo

"J- Z (Yi/#i)`+l Z ~id°i(k) "q- Z Yh Z ( Y i / . i ) I Z dhi(k) l=1 k = t + l h = l 1=1 k=l

o o o o ¢K)

= Yi- ;idoi(O)+(Yi/#i)Ai Z doi(k) +Z(Y,/#i) ̀+' Z aid°i(k) k = 2 l=1 k = l + 2

q- Z Yh Z ( ~ ' / # i ) I dhi(k) -ah i ( l ) -- Z Yhdhi(O) h = l l=1 h = l

N o0 0o

= Yt- .X,do,(O)- y ~ rhdh,(O) + ~(YdU,) ~ Y~ Aido,(k) h=l l=l k=l+l

N oo oo

+ E2 r~ E2(r,/,,)' X2 d~,(~). [] h = l /=1 k = l + l

5 0 0 W. Henderson et al./Geometric equilibrium distributions

L E M M A 3.2

N N

i=1 i=l

Proof

From lemma 3.1,

N N N N N

i=1 i=1 i=l i=1 h = l

N o~ oG

i=1 1=1 k = l + l

N oc~ N oo

+ Z Z(~,/.;/'Z ~ Z d~,(k/ i=1 /=1 h = l k = / + l

N N o~z o c N

= - Z / ~ i d ° i ( O ) q - Z Z (Yi/#i)l)~i Z doi(k) q - Z Yi i=1

N

+ Z h = l

N

=-y~ i=1

N

+ Z h = 1

N

- Z h = l

N =-y~'

i=1

N

+ Z h = l

i=1 /=1 k = / + l i=1

N oo co N N

i=1 /=1 k = l + l h = l i=1

N oo oo N

/~id°i(O)~- Z Z (Yt'/~i)`/~i Z doi(k)-~ Z ri i=1 l=1 k = / + l i=1

N c~ c~ N N

i=1 /=1 k=l h = l i=1

N oc

i= I k = l

N oo oo

;,e0,(0)+~ ~(r,/.,)'a, ~ ao;(k) i=1 1=1 k = l + l

N oo oo

Yh Z Z ( Y i / # i ) / - I Z dhi(k) by eq. (l). i=1 /=1 k=l

[]

W. Henderson et aL/Geometric equilibrium distributions 501

LEMMA 3.3

{Y/, i = 1 , 2 , . . . , N } satisfy

Aj= Bj,

iff they also satisfy

j~S j E S VSC_ {1 ,2 , - . . ,N} , (6)

A i - b B i = O , Vi E { 1 , 2 , . . . , N } . (7)

Proof

Let S1 = SzU {i} SO_ {1,2, . . . ,N}. Then

and therefore

for s o m e i ~ S 2. Assume eq. (6) holds for all

j¢SI jESt

Z A j - A i = Z Oj-~-n i. (8) j¢$2 jES2

Since eq. (6) holds for $2 we immediately obtain eq. (7) for a particular i. We can repeat this for all i E { 1,2,- . - , N }. That is, eq. (6) =¢, eq. (7). For the converse we use induction.

Assume that we have found { Y/i = 1,2, . -- , N } which satisfy eq. (7) for all i = 1 ,2 , - - . ,N and also satisfy eq. (6) for all sets S1 C_ { 1 , 2 , . . . , N } : ]Sl] > k for same k > 0. Choose a particular $2 : 1S2] = k and consider eq. (8) again. In parti- cular, observe that eq. (7) holds for all i E {1,2, . . . ,N} and with $1 = $2 U {i} eq. (6) holds for $1.

j~-S1 jESI

j¢$2 jES2

502 IV. Henderson et al./Geometric equilibrium distributions

i.e. eq. (6) must hold for any choice of $2 and/such that Is21 = k. Thus we have that eq. (6) holds for all S C { 1 ,2 , . . . , N } : IS l > k, establishing the induction argument and we now need only a starting point to complete the proof.

Consider the case IS I = N, that is the set S = { 1 ,2 , . . . , N }. We need to show that eq. (6) can be obtained from eqs. (7) for this particular choice of S, that is, we need to show that E~'= I Bi = 0. Assume eq. (7) holds, then

( r , / m ) (A; + B,) = 0.

Summing over i and using lemma 3.2 we have

N N

+ B,) = B, = O, i=1 i=1

i.e. eq. (6) holds for S = { 1,2,--. , N }, thus completing the induction. []

T H E O R E M 3.1

For the queueing network described in section 2, eq. (3) is an invariant measure iff there exists a non-negative solution, { Y,. }, to the following non-linear system of equations

= [ £ ] )~idoi(O) + Yhdhi(O) h = l

- Z ( Y i / # i ) l doi(k) + Yh dhi(k) , 1=1 1 h = l k = l + l

(9)

for / = l , 2 , - - . ,N .

Proof

Equations (5) and (6) are equivalent and lemma 3.1 indicates that eq. (9) is equivalent to (Yi/#i)(Ai + Bi) = 0. Thus lemma 3.3 verifies that the N eqs. (9) can hold iff the 2 N eqs. (5) also hold, i.e. { Yf, i = 1 ,2 , . . . , N } satisfy eqs. (5), which are the global balance equations when multiplied by p(n) from eq. (3). []

Remark 3.1

It is known from the theory of Markov processes [7] that a regular irreducible Markov process has a unique invariant measure up to constant multiples implying that if there exists a solution { Yi } to eqs. (9) then it is unique. Our experience shows that for arbitrary parameter values repeated substitution in eq. (9) leads to a unique solution. []

IV. Henderson et al./Geometric equilibrium distributions 503

In the case where eq. (3) is an invariant measure for the network, and Y,. < #i for i = 1,2, .- . ,N ensuring summability, we set K = II~l(1 -(Yi/izi)) in eq. (3), making it the equilibrium distribution of the network and the following interpreta- tions may be made

(Yi/#i) l = Pr(queue i contains at least I customers),

(1 - (Yi/#i)) (Yi/#i) t = Pr(queue i contains exactly l customers).

Remark 3.2

The parameter Y/has an interpretation as the throughput of the single server at queue i. []

Consider the balance required between the rate at which single positive cus- tomers enter queue i (including those which are immediately eliminated by their own negative customers) and the rate at which customers leave queue i, either due to a service completion, or after being cancelled by the arrival of a negative customer. This balance is given by

N oo

j = l k=O

= Yi+Ai l (Y i /# i ) ldo i (1)+Zl (1- (Y~ . /# i ) ) (Y i /# i ) 1-1 doi(k) /=1 /=1 k=l

+ Z YJ l(Yi/#i)tdyi(l)+ l (1-(Yi /#i))(Yi / l~i) ' - ' dji(k) j = l 1=1 l=1 k = l

(lO)

Equation (10) was obtained directly by interpreting the parameter Y,. to be the throughput of the single server at queue i, i.e. it is a logical traffic equation describ- ing the relationship between the throughputs of the servers in the network. We now show that it is equivalent to eq. (9) which was obtained as the condition under which the network has a geometric invariant measure. The RHS of eq. (10) consists of terms describing the rate at which customers leave queue i. This can occur through service completions (the Y~ term), or due to the effects of negative customers on queue i. A batch of negative customers can be created at queue i only following the arrival of a positive customer to queue i. The first summation term inside each of the two square brackets describes instances in which batches of l negative customers are generated, but do not empty the queue. The second summation term in each set of brackets accounts for all transitions in which queue i is emptied due to the generation of at least l negative customers (remembering that the excess

504 W. Henderson et al./Geometric equilibrium distributions

negative customers have no effect on queue i once it has been emptied). The LHS of eq. (10) consists of all positive customer arrivals to queue i, either from outside of the system, or as that proportion of the throughput of queue j which is transferred to queue i. We consider a positive customer to have arrived at the queue before any batch of negative customers can be generated.

We can simplify each of the terms in square brackets on the RHS ofeq. (10). If we let Z / = (Yi/#i) and consider the outside of the system to be queue 0, then for j = 0 , 1 , . . . , N ,

~ c o c ~ c

lZ[dj,(l) + ~ I(1 - Zi)Z[-' ~-~ dj,(k) 1=I 1=1 k=l zc vc k - I

l z / 4 , ( z ) + (1 - z,) Z dt,(k) ~ (t + 1)z~ /=1 k = l 1 = 0

= ~ IZ[4i(l ) + (1 - Z,) dii(k) "~i Zi /=1 k = l

I- / -I

' d I Z , - Z r + , l = ~l=, lZ[dji(l)-4-(1 - Zi)k=lZdji(k)~ i - - [ -1--~'Z "J

oc ,

tz~4,(t) + - - / = 1

OG

1 k~ 1 dyi(k)[1-(k + 1)Z~ + k Z f+l] 1 - Zi .=

0(5

- l - z i ~ dj;(t)[1- z~] / = 1

cc 1 - 1

1=1 k = 0

vc l

dji(l + I) Z(Y i /# i ) k. l = 0 k = 0

Therefore

N ~c ~c k

~,+ ~ ~ 4,(k/= r,+ ~ ,~ do,(~ + 11 ~(r,/p,/t j = l k = 0 k = 0 l = 0

N oo k 4- ~ Yj ~-~ dji(k + 1)~"~(Yi/Pi)'.

j = l k = 0 / = 0

W. Henderson et al./Geometric equilibrium distributions 505

Thus, by eqs. (1) and (2),

Yi=Ai 1 - d o i ( k + l ) (Yi/#i)/ + Z ~ ' Z d j i (k) -d j i (k+l)Z(Yi /# i ) l k = 0 = j = l k = 0 1=0

= Aidoi(O)+ Yjdji(O)-Ai doi(k+l)Z(Yi /# i ) I j = l k = l /=1

N ~ k

- G Z d;,(k + l) (y,/.,)' j = l k = t / = l

= a ;do , (0 ) + b d ; ; ( o ) - ( g / m ) t j = l = 1

N

do,(k) + ~ rj Z 4 , ( k ) j = l k = l + l

which is eq. (9). That is, we have derived eq. (9) from a logical traffic equation for throughputs.

4. Partial balance

With a geometric equilibrium distribution there is always a connection between traffic equations and partial balance equations, found by multiplying or dividing by p(n). In this section we derive these partial balance equations for this network. These balance equations have natural interpretations, and are related to the effect that the boundaries of the state space have on the system.

THEOREM 4.1

Equation (3) satisfies the global balance eqs. (4) iffit satisfies the N + 1 partial balance equations

N

p(m - e i )A ido i (O ) + Z p(m - ei + eh)lZhdhi(O) h = 1

oG ¢(~

=p(m)#i+ Z p ( m + ( l - 1 ) e i ) Z Aid°i(k) / = l k = t + l

N oo oo

-4- y~ ~h ~_, p(m + (1- 1)el + eh) ~ dhi(k), h = l / = l k = l + l

( l l )

506 W. Henderson et al./Geometric equilibrium distributions

for i = 1,2, . - . ,N and

N N o o o c

p(m) ~ A,d0,(0)= ~ A,~--'~p(m + le,) ~_~ doi(k ) i=1 i=1 1=1 k = l + l

N N oo oo

+~--~-~#h~-'~p(rn+lei+eh) ~ dki(k ). (t2) i=l h = l 1=0 k = l + l

Proof

Theorem 3.1 gives the result that { Yi, i = 1,2,..-, N } satisfy eq. (9) which, when multiplied by (#i/Yi), becomes

• ] N o o o o

Izi- (I-zi/Yi) Aidoi(O) - ~_ a Yhdhi(O) q-~'~(Yi/#i)/-1 ~ " ~ Aidoi(k ) h = l I=l k = / + l

N o o o o

+ Z Yh ~-~(Yil#i) l-' ~ dhi(k) = 0 . h = l l=1 k = l + l

Multiplying this in turn by p(m) we obtain

N o o o c

p(m) # , - (#,/Y,.)(Afloi(O) - ~ Yhdm(O)) + ~(Y,-/#i) ' - ' ~ A,d0,(k) h = l t= t k = l + l

-~- ~ Yh Z (Y i /~ i ) t-1 ~ dhi(k) J h= l 1=1 k = l + l

= 0

N

p(m)#i - p ( m - ei) Aidoi(O ) - ~ p(m-- ei + eh)#hdhi(O) h = l

o o (3o

+ ~"~p(tn+ ( l - 1)ei) ~ Aidoi(k ) /=1 k = l + l

N o o o o

-3 V ~--~h ~-~p(tll-3v(l - 1)ei-3veh) ~ dhi(k) =0, h = l 1=1 k = l + l

(13)

which, with some reordering, is the partial balance eq. (11). We now use this result

W. Henderson et al./Geometric equilibrium distributions 507

to derive partial balance eq. (12). Rewrite eq. (4) so that all non-zero terms are on one side

P(m)[i~CsAi(1-d°i(1))+Y~Aid°i(O)+Y~#i]ics i¢s -Y~P(m-ei)Aid°i(O)i¢s oo oc oo

-Y~_~p(m+lei)Aidoi(l+ 1)--~--~p(m+lei) iq~S 1=1 iES l=1 k = l + l

/~idoi(k)

-~-'~#h P(m+eh+(l-1)ei)dl, i(l) h=l L i~s t=0

- ~ f ~ p ( m + e h + (l-- 1)ei)~ din(k) =0. i E S t=1 k = l

(14)

Sum eq. (13) over i ¢ S for S C {1 ,2 , - . . , N } , and subtract from eq. (14) for the same S to obtain

0 =p(m) [

i¢s

iCS

Ai(1 -- doi(1)) q- ~ Aidoi(O ) i•S i E S

~_, p(m+ lei)Ai y~ doi(k ) - y~.p(m)Ai~_, doi(k) l = l k = / + l iriS k = 2

N oo 00

Y~ #h Y~ p(m + (l-- 1)ei + eh) ~ dj, i(k) h = l I=1 k = l

N o~ oG

~-~ ~-~#h ~ p(m + (l-1)ei + eh) ~ dhi(k) iES h = l l=1 k = l

i f fS l=1 k = l + l

N N o c o c

i=1 i=1 l=1 k = l + l

N N oe

-~-~-~#h~_.~P(m+lei+eh) ~ dhi(k), i=1 h = l / = 0 k = l + l

doi(k)

which is equivalent to eq. (12). Note that eq. (12) is independent of S.

508 W. Henderson et al./Geometric equilibrium distributions

To complete the proof, we need to be able to derive the global balance eqs. (4) for an arbitrary S c_ {1 ,2 , . . . , N} from eqs. (11) and (12), assuming that the invari- ant measure is given by eq. (3). This is achieved by using the same symbolic notation (Ai's and Bi's) as defined in section 3.

N

eq. (12),: ',, p(m) Z Bi = O; i=1

eq. (11)< >p(m)(A i+Bi)=O.

Thus for a given set S c { 1 , 2 , . . . , N }

Z (eq. ( 1 1 ) ) - e q . (12) = 0 i ¢S

N

Z p(ra)(Ai+ Bi)-p(m ) Z B i = O i f S i=l

-: > eq. (4). []

Referring to eqs. (I 1) and (12) as partial balance equations is not strictly accurate. Partial balance equations usually correspond to a simple separation of the terms of the global balance equations and are equivalent to considering global balance equations on an isolated subset of the sample space. This is not the case for the models of Gelenbe et al. [I-3], nor for the model presented herein. The balance observed in each of these works is closer to, but not identical to, the equations derived by considering balance in and out of subsets of the state space.

Most partial balance concepts are clearer when explained in terms of prob- ability flux which is relatively easy when queues only have positive customers (see, for example, Whittle [8]). In our case it is a little more difficult but it does give some feel for the internal balance in this network and may provide sufficient insight to yield further generalisations.

Equations (11) and (12) are not independent. Equation (12) can be derived from eqs. (11) by replacing m - e i with m and then summing over i. Conse- quently, an interpretation of these partial balance equations as flux equations is necessary only for eqs. (11).

The left hand side of eq. (11) is the flux into state m due to the transfer of a positive customer into queue i. The terms on the right hand side of eq. (11) are respectively the flux out of state m due to a departure from queue i and the flux due to the arrival of any type of customer to queue i which leaves all other queues

W. Henderson et al./Geometric equilibrium distributions 509

in state mj, j ¢ i, and at the same time decreases queue i from at least m i to less than m i customers. This flux balance is not as simple to explain in a few words as the flux balance for Jackson networks. The best approach that we can offer is via the con- cept of "source i interim states" which are N - 1 vectors of the queue lengths observed, at queues other than queue i, by any customer in transit to or from queue i. To this end define, for each m E Z N, m(i) E Z Iv-1 as re ( i )= (ml, m 2 , . . . , m i - l , mi+l, • . . ,mN).

For an arbitrary state m E S, and for a fixed i E { 1 , 2 , . . . , N }, eq. (11) can be expressed as

the probability flux balances between the sets of states {n :n i < mi} and {n : n i > mi} whenever the source i interim states have the property n(i) = re(i).

When there are no negative customers in the network this flux balance becomes the standard Whittle [8] balance for Jackson networks relating flux into state m due to an arrival to queue i to the flux out of m due to a departure from queue i.

5. Gelenbe's results

In this section we show that the geometric product form equilibrium distribu- tion derived by Gelenbe [1] for a network of queues with movements of single posi- tive and negative customers can be derived from theorem 3.1 as a special case.

Gelenbe considered an open network of N single server queues with mutually independent, iid exponential service time distributions of rates r ( 1 ),. • •, r (N). Posi- tive and negative customers arrive to queue i according to independent Poisson pro- cesses of rate A(i) and A(i), respectively. After completing its service at queue i, a positive customer will transfer to queue j as a positive customer with probability p+(i, j ) , as a negative customer with probability p-( i , j ) , or it will depart the sys- tem with probability d(i). As in our system, a customer is not allowed to transfer to the queue at which it was served. Gelenbe showed that the geometric product form equilibrium distribution for this system exists with the following form

N A+(i) (15) p ( n ) = l - ~ ( l - q i ) q 7 ~, w h e r e q i - r ( i ) + A- ( i ) '

i=1

iff there exist unique non-negative A+(i) and A-(i) for i = 1 , - . . , N that satisfy the following system of non-linear simultaneous equations

N N

A + ( i ) = ~ q j r ( j ) p + ( j , i ) + A ( i ) , A - ( i ) = y ~ q j r ( j ) p - ( j , i ) + A ( i ) . (16) j = l j = l

510 W. Henderson et al./Geometric equilibrium distributions

This system can be demonstrated to be a special case of our system by setting a few parameters. From our notation we require

~i = r ( i ) , A i = A ( i ) + A ( i ) ,

A(i) A(i) d°i(O) = A ( i ) + A ( i ) ' d°i(1) = 0, doi(2 ) - A(i) + A ( i ) '

dij(O) = p + ( i , j ) , dij(1 ) = d(i) dij(2 ) = p - ( i , j ) , N '

dij(k) = 0, for any other combinations of i, j, k,

A+(i) (Yi/#i) = qi - r(i) + A-(i) "

NB: Here we have arbitrarily set dij(1) = d( i ) /N , since the transition it describes has no net effect on queue j (a customer transfers from queue i to queue j and then leaves the system as a negative customer). We only require that EU=l dq(1) = di. Then our equilibrium distribution is identical to that of Gelenbe if we can obtain the logical flow equations for Y/, (i.e. eq. (9), from eqs. (16)).

A+(i) (Yi/I-ti) = qi --

r(i) + A-(i)

(~--~=N= 1 q j r ( j ) p+ (j , i) + A(i))

# i + ~-~'~N=I q j r ( j ) p - ( j , i ) + A(i)

aj;(0) + A;do;(0) #i + )--~Ul Yjdj/(2) + do/(2)A i"

Multiplying by the RHS denominator, and isolating the resulting Yi term, we obtain

Yi = Z Yjdji(O) + Aidoi(O) - (Yi/]zi) Yjdj i(2) + A;doi (2 ) j = l =

N Yjdji(O) + Aidoi(O) (Yi/lzi)' Z YJ Z dji(k) + Ai Z d°i(k) = = j = l k = l + , k = l + ,

(since dji(k) = 0 for k > 2), which is equivalent to eq. (9). Gelenbe and Schassberger [3] proved the existence and uniqueness of a solu-

tion to eqs. (16), which are equivalent to the throughput equations for a system in which only single negative customers are routed through the networks. The adapta- tion of this proof is straightforward to the case of batch routing of negative custo- mers, thus ensuring that a unique solution exists to eqs. (9).

IV. Henderson et al./Geometric equilibrium distributions 511

6. Conclusions

In this paper we have extended the set of queueing networks that include negative customers and have a geometric product form equilibrium distribution. We showed that the global balance equations for these networks can be built up from a number of partial balance equations relating to the effect of the boundaries of the state space on the network, and that the partial balance equations are equiva- lent to a set of logical traffic equations.

We are confident that we can further extend the class of product form queue- ing networks incorporating negative customers, by generalising the form taken by the partial balance equations of these current networks. We also intend to investi- gate reversibility for networks with negative customers, and possibly introduce state dependent equilibrium distributions to be able to analyse networks of queues with batch arrivals of positive and negative customers.

Acknowledgement

The authors would like to thank the Teletraffic Research Centre at the Uni- versity of Adelaide and Telecom Australia for supporting this research, and acknowledge the permission of the Director General of Telecom Australia to pub- lish this research. We wish also to thank Nico van Dijk for introducing this problem to us, and the referees for their constructive comments.

Addendum

It has come to our attention, since the original submission of this paper, that Gelenbe [4] has also shown that networks of queues with batch routing of negative customers have a geometric product form distribution. Gelenbe's model is different from ours in that it assumes that an arriving customer removes a batch of customers from queue i with a probability distribution that does not depend on its node of origin. In contrast, our model as described in section 2 allows the distribution of the number of customers removed to depend on both the origin and destination queue of the moving customer.

References

[1] E. Gelenbe, Product form networks with negative and positive customers, J. Appl. Prob. 28 (1991) 656-663.

[2] E. Gelenbe, P. Glynn and K. Sigrnan, Queues with negative arrivals, J. Appl. Prob. 28 (t 991) 245-250. [3] E. Gelenbe and R. Schassberger, Note on the stability of G-networks, Research Report 90-8,

Universit6 Ren6 Descartes, Paris (1990). [4] E. Gelenbe, Negative customers with batch removal, private communication. [5] J. Jackson, Networks of waiting lines, Oper. Res. 5 (1957) 518-521. [6] E. Kandel and J. Schwartz, Principles of Neural Science (Elsevier, Amsterdam, 1985). [7] S.M. Ross, Introduction to Probability Models, 4th ed. (Academic Press, 1989). [8] P. Whittle, Equilibrium distributions for an open migration process, J. Appl. Prob. 5 (1968) 567-571.