Multiclass Queueing Networks with State-Dependent Routing

125

Multiclass Queueing Networks with State-Dependent Routing A.E. K r z e s i n s k i * Institute for Applied Computer Science, * * Unioersity of Stellenbosch, 7600 Stellenbosch, South Africa Received 10 February 1983 Revised 9 May 1984 and 7 May 1986

A Multiclass Queueing Network (MQN) Q(N, ~¢[) consisting of M BCMP centers with index set Jr' and population vector N is partitioned into two subnetworks Q(N - V, ,¢t - ~ ) and Q( v, ~"). The centers in Q( v, ~ ) are partitioned into disjoint groups called branches. The branches are arranged into a hierarchy of nested subnetworks. A set of State-Dependent Routing (SDR) probabilities is used to admit customers from Q ( N - V, ~ - ~ ) into the individual branches of Q( v, ~e'). The SDR probabilities are products of linear functions of the branch and subnetwork populations such that Q(N, .,¢/) has a product-form solution. The SDR probabilities set an upper bound on the number of customers that can concurrently be present in each branch and each subnetwork of Q(V, ~ ) . Both convolution and Mean Value Analysis (MVA) methods can be used to compute the performance measures for a MQN containing both state-dependent and state-independent routing. A summary of the SDR MVA solution method is presented. Finally, the effects of SDR on network performance measures are investigated.

Categories and Subject Descriptions: D.4.4 (Operating Systems): Communications Management - network communication; D.4.8 (Operating Systems): Performance - modelling and prediction; queueing theory.

Keywords: Adaptive Routing, Convolution Algorithm, Load Ba!ancing, Mean Value Analysis, Multiclass Queueing Networks, Population Constraints, Product-Form Solutions, Product-Form State-Dependent Routing.

1. Introduction

Large computer systems and computer networks often have a sufficiently large and interconnected set of resources to offer alternate processing sites for individual service requests. An adaptive routing strategy can be used to select processing sites and assign them to service requests in such a way as to minimize the queueing delay at each processing site. The impact of adaptive or State-Dependent Routing (SDR) on system performance is not intuitively obvious. Methods need to be developed to evaluate the performance benefits of SDR in terms of the arrival rates and service requirements of the various workloads and the service rates and interconnections among the alternate processing sites. For example, given several identical processing sites each with a * Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. ** The Institute for Applied Computer Science is sponsored by Reunert Computers (Pty) Ltd. ..........

Anthony E. Krzeslnsid obtained the M.Sc. degree in Theoretical Physics from the University of Cape Town in 1968 and the Ph.D. degree in Physics from Cambridge University in 1971. After completing this doctorate he joined the Mathematics Department of the Shell Laboratory, Amsterdam in 1972. In 1975 he joined the Department of Computer Science at the University of Stellenbosch, South Africa. In 1981 he cofounded the Institute for Applied Computer Science at the University of SteUenbosch. He was appointed Professor of Computer Science at the University of Stellenbosch in 1985. In 1982 he was a Visiting Scientist at the IBM TJ Watson Research Center, Yorktown Heights, New York. His research interests concern exact and approximate analytic solution methods for multiclass queueing networks and the application of queueing network models to evaluate the performance of computer systems.

North-Holland Performance Evaluation 7 (1987) 125-143 0166-5316/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)

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A.E. Krzesinski / Multiclassqueueingnetworks

distinct waiting line, the optimal routing policy is to assign a task to the processor with the shortest waiting line. However, when the sites have unequal processing rates, a problem arises as how to combine the benefits of load balancing while preferentially routing tasks to the faster processors. This paper addresses a particular type of SDR and a particular interconnection of processing sites which together allow S D R to be described in terms of a Multiclass Queueing Network (MQN) with a product-form solution. These SDR functions permit a larger variety of state-dependent behavior than the S D R functions presented in [10]. Specifically: - SDR is extended from single-class to multiclass queueing networks, a - the relationship between SDR and the imposition of population bounds on queues and on groups of queues is investigated, - the relationship between SDR and a form of blocking is investigated, - a M Q N with S D R is shown to be equivalent to a M Q N with state-independent routing whose centers have certain state-dependent service rates, - MVA methods are presented for computing the performance measures of a M Q N with SDR. The paper assumes the convolution results [2,7] for M Q N s of BCMP centers [1] for computing the network normalizing constant and the performance measures. The MVA results [8,9] for M Q N s are also assumed. Section 2 presents the product-form S D R probabilities. The consequences of SDR, namely adaptive routing and the imposition of population bounds on queues and upon groups of queues are discussed. Section 3 presents the joint probability distribution for a closed M Q N with SDR. Section 4 summarizes the MVA solution of a M Q N with SDR. Section 5 presents a numerical investigation of the properties of SDR. Appendix A presents the derivation of a product-form expression for the joint probability distribution for M Q N s with SDR.

2. S t a t e - d e p e n d e n t routing

2.1. Definitions Let Q(N, ,At) denote a M Q N consisting of M BCMP service centers with index set ~¢/= {1 . . . . . M}. Let the customers belong to J closed chains labelled (1 . . . . . J ) . Let N = ( N 1. . . . . Nj) denote the population vector where Nj is the number of chain j customers. Let nij denote the number of chain j customers at center i and let n i = (n n + • • • +nij) denote the total number of customers at center i. The network state descriptor is given by N = ( n 1 . . . . . riM) where the population vector at center i is given by n i = (nil . . . . . nij ). Let lj denote a unit vector in the j direction. Let the set of center indices ~ be partitioned into two subsets J r ' - ~ and ~" such that the subnetwork Q(V, ~tr) contains all the centers which are subject to SDR. Let Q(V, ~¢r) have a single entry center labelled e and a single departure center labelled d. Note that the centers e and d are not in

Q(V, ~ ) . Let ..g also be partitioned into B subsets ~ a . . . . . ~B where ~ ¢ - Y/'= ~1 and Y/'= ~ 2 "-b • • • " [ - ~ B " The subnetworks Q(B b, ~b) are referred to as branches. Each branch Q(Bb, ~b) where 2 ~ b ~< B has a single entry center e(b) directly connected to e and a single departure center d(b) directly connected to d. Fig. 1 presents a schematic representation of a branch Q(B b, ~b). The centers in the branch (a few of the centers are represented in the figure) are arbitrarily interconnected with the restriction that all customers enter the branch via a single entry center and depart from the branch via a single departure center. Fig. 2 illustrates a M Q N Q(N, ..g) consisting of four branches. The M Q N Q(N, ~¢¢) is partitioned into two subnetworks Q ( N - V, ~¢g- ~e') and Q(V, Y/'). The subnetwork Q(V, ~t") has a single entry center with index e and a single departure center with index d. The entry center e is directly connected to the entry centers of the branches in Q(V, ~e'). The departure centers of the branches in Q(V, Y/') are directly connected to the departure center d. a Reference [10] refers to a dissertation where multiclass SDR results are presented.

A.E. Krzesinski / Multiclass queueing networks

127

© Fig. 1. The branch B(Bb, ~b).

The type of S D R probabilities considered in this paper introduces a hierarchical structure on the branches. Define a sequence of sets { 5 / 1. . . . . 5/'r } of center indices such that ft = 5:r+ 1 c 5 : 7. c 5 : r - 1 c • • • c 5:1 = 5:- The sets ~tt are chosen such that for any b r a n c h index b, 2 ~< b ~< B, there exists a unique subnetwork index t, 1 ~< t ~< T, such that ~ b C 5:t -- 5:t+1. Each branch is thus completely contained in the difference of two hierarchically adjacent subnetworks. F o r example, consider ~ b C ~ t -- ~ t + l • Each center in Q(Bb, "~b) is in Q ( V t, Y/t) but not in Q(Vt+ 1, ~ t + l ) and Q(B b, ~b) belongs to a total of t nested subnetworks Q(V1, 5:1) . . . . . Q(Vt, Y'~t)Fig. 3 illustrates how the subnetwork Q(V, 5 : ) is partitioned into (in this case) two nested subnetworks Q(V1, 5:1) and Q(V2, 5:2) such that f[ = 5:3 c Y:2 c 5:1 = 5 : where 5:1 = ~ 2 + ~3 + ~ 4 and 5:2 = ~3. Each b r a n c h is contained in the difference of two hierarchically adjacent subnetworks. Thus 5:1 - 5:2 = ~ 2 + ~ 4 and 5/2 - Y:3 = ~ 3 . Let mbj denote the n u m b e r of chain j customers in b r a n c h b and let m b denote the total n u m b e r of customers in b r a n c h b. Then J

mbj= ~, nq i ~ ,~'t,

and

mb= ~_, mbj. j=l

t/

:ZD5D

5¢3B

Fig. 2. A queueing network consisting of four branches.

A.E.Krzesinski/ Multiclassqueueingnetworks

128

gl

SE) ©

) B5

$33 ZD ~ )

/3

Fig. 3. Nested subnetworks.

Let ~¢t denote the index set of the branches in Q ( Vt, ~e~t). Let vtj denote the number of chain j customers in subnetwork t and let vt denote the total number of customers in subnetwork t. Then J

otj= E nij~ E mbj i~

and

vt= E Orj"

be.~ t

j=l

2.2. The state-dependent routing probabilities Consider a M Q N whose centers are arranged into a hierarchy of nested subnetworks as described in Section 2.1 above. The following State-Dependent Routing (SDR) and State-Independent Routing (SIR) probabilities are used to describe the routing of the customers: (i) from the entry center e of Q(V, YP) to the entry centers of the various branches in Q(V, ~"), (ii) among the centers within each branch and eventually to the departure center d from Q(V, ~e'), (iii) from the departure center d to the centres of Q ( N - V, ..¢,[- ~¢") and ultimately to the entry center e of

Q(V, r').

The remainder of this section describes the routes (i), (ii), and (iii) in detail. (i) The SDR probability of a chain j customer, upon completing service at the entry center e of Q(V, ~'), proceeding directly to the entry center e(b) of branch b where B b c ~ - ~¢'~t+1 and 2 ~
Pej,e(b)J(N)

=

( 6,vj(rnbj ) sO1 ('Os-lsj(OsJ) O~ssj(Osj) =

0

if

OOssj(Osj) >

0 for all 1 4 s 4 t,

(1)

if %sj (osj) = 0 for any 1 ~< s ~< t.

The functions ~0 are defined to be nonnegative and hence the function 8 is nonnegative. The functional forms of 8 and ~0 are derived in Section 2.3 below.

A.E. Krzesinski / Multiclass queueing networks

129

(ii) A set of SIR probabilities Pxj,yj where x and y are in ~b and 2 ~< b ~< B is used to route the chain j customers among the centers in branch b. The SIR probability Pd(b)j,aj = 1 is used to route customers from the departure center d(b) of branch b to the departure center d of Q(V, Y/'). Transitions from one branch b to another branch f where 2 ~< b, f~< B are not permitted. (iii) A set of SIR probabilities P~j,yj where x and y are in J r ' - z e " is used to route the chain j customers among the centers in Q ( N - V, : g - Y/'). It is shown in Appendix A that the global balance, equations for a M Q N with S D R and SIR probabilities as described above have a product-form solution. S D R probabilities of the type presented in (1) above are therefore sufficient for a product-form solution. It remains to be shown that equation (1) is necessary for the existence of product-form solutions. S D R probabilities other than equation (1) can exist which also yield product-form solutions. However, in the context of this paper a S D R probability will refer to a probability of the type defined in equation (1).

2.3. A functional form for the state-dependent routing probabilities In order for equation (1) to define a proper probability distribution it is necessary that for each chain index j, 1. ~ j ~ J, B

T

S = Pej,dj(N) + E Pes,~(b)j(N) = P~j,d/(N) + E b=2

t=l

~-,

Pej,e(b)j(N) = 1,

(2)

b ~,J~t -,.~¢t + 1

where Pej.aj(N) defines a routing path connecting the entry center e and the departure center d of Q(V, ~P). Case 1. Let K be the largest integer 1 ~< K~< T such that %~j(vsj ) > 0 for all 1 ~< s ~< K. Substituting (1) into (2) yields

S = eej.dj(N) + E

H

t=l s=l

= eej,,j(g) ,,

+

¢Ossj(Osj)

E

8tba( m bj )

b ~a~et_ jaet+1

O'llj(V.)

,-1

+ Et :F2 Is : l

~ssj(Vsj)

OOttj(Vtj)

E ~tbj(mbj)-- E ~,-lbj(mbj)] • b~s4, beat,

(3)

Substituting .,,j(v,+) =

E

(4)

1 < t <. K ,

and

°at-ltj(vO) = E 8t-,bj(mbj), b ~a¢,

1 < t <~K,

(5)

into equation (3) yields S = Pej.aj(N)+ Oaol/(vlj ). The identity S = 1 can be realized by assigning

Pej,dj( N) = 1 - ¢O01j(Vlj ). The remainder of this analysis assumes that if ~on/(Vlj ) > 0 then ~oOl/(Vlj ) = 1 and Pej,aj(N) = O. Case 2. If % l j ( v a j ) = 0, then we have S = Pe/,di(N). The identity S = 1 can be realized by assigning

Pej,dj(N) = 1. The functional forms of 6 and ~: are established as follows. The linear equations (4) and (5) together with the linear identity v,j = Y.b ~ ~,,mbj imply that 8,bj(mb+ ) = Ctjmbj + dtb+

and

~t_lbj(mbj) =

C,_,jmbj + dt_aa/,

(6)

A.E. Krzesinski / Multiclass queueing networks

130

where Cu, Ct_ u, dtbj and dt_lb j

constants and ~b C ~ -- ~+1- Substituting (6) into (4) and (5) yields

are

t'Ottj(Vtj ) = Ctjvtj + Dttj,

1 <~t <~T,

(7)

£Ot--ltj( Vtj ) = Ct_ljVtj + Dt_ltj,

1 < t <~T,

(8)

where

Dttj= ~ dtbj f o r l ~ < t ~ < T

and

Dt_~tj= ~ d,_lbj

be~¢,

forl
(9)

be~¢,

2.4. Chain-independent state-dependent routing The remainder of the analysis assumes that the SDR probabilities are chain independent. The chain-dependent analysis is presented in [5]. In the case of chain-independent SDR, the probability that a customer, upon completing service at the entry center e of Q(V, ~ ) , proceeds directly to the entry center e(b) of branch b where ~b C ~ -- ~,+1 and 2 ~
Pe,e
=

$tb(mb) s1--I Ws,(V~) =l

if o~ss( vs) > 0 for all 1 ~
0

if %~(v~) = 0 for any 1 ~
(10)

Equations (6) through (9) now become

~tb(mb) ~-- CtTYIb -~- dtb

and

(11)

~t_lb(mb) = Ct_lm b + dt_lb,

where C. C,_ 1, dtb, and dt_lb are constants and ~ b Q ~ t -- ~ t + l , and

%,(o,)=C,v,+D., ¢o,_,,(v,)=C,_lvt+D,_a, ,

l
D , = E dtb for 1 ~
and

(12) (13)

where

Dr-l, = E dt-lb be ~

for 1 < t ~< T.

(14)

Note that the SDR probabilities (10) can be reduced to a SIR form by setting the Ct = 0 yielding t

Pe,e(b) = dtb 1-I Os- ls/Oss.

(15)

s=l

2.5. Adaptive routing, concurrency constraints and blocking The assignment of positive values to the dtb and negative values to the C, [10] allows the SDR probabilities to preferentially route customers to the least congested branches in Q(V, Y/'). T h e SDR probabilities also set upper bounds on the population within each branch Q(Bb, ~b) and within each subnetwork O(V, ~t)For example, consider a single-class closed queueing network Q(N, ~¢[) consisting of a central server and two peripheral servers. Let ~¢--- (1, 2, 3}, ~ = ~1 = {2, 3}, and ~z = {3}. Let each center define a single branch so that ~1 = {1}, ~2 = (2}, and ~3 = {3}. The central server center 1 functions as both the entry center and the departure center for Q(V, ~ ) . Let C1 = C2 = - 1 . Let (d12, d13) = (1, 2). Let d23 = 3. Equation (14) yields the coefficients DH = d12 + d13 = 3, D12 --- d13 = 2, and D22 = d23 = 3. Equation (11) yields the branch population constraints: (~12(m2)

=

1-

m 2 >/0

SO that m 2 ~< 1

and

(~23(m3)

=

3-

m 3 >~ 0

SO that m 3 ~< 3.

A.E. Krzesinski / Multiclass queueing networks

131

Table 1 m2

m3

el

02

P1.2

P1,3 2

P1A

0

0

0

0

~

~

0

0 1 0 1 1

1 0 2 1 2

1 1 2 2 3

1 0 2 1 2

½ 0 1 0 0

~ 1 0 1 0

0 0 0 0 1

Equation (12) yields the subnetwork population constraints: ~ 0 1 a ( V l ) = 3 - v 1>/0

so that v1~<3

and

o ~ 2 2 ( v 2 ) = 3 - o 2 >/0

so that v2~<3.

Equation (13) yields a tighter bound on v2, namely ~ 1 2 ( v 2 ) = 2 - v 2>/0

so that o2~<2.

Note that in the above example the bound on vl (01 <~ 3) is less than the sum of the bounds on m 2 (m 2 ~< 1) and m 3 ( m 3 ~< 3) even though Y/'I = ~2 + ~3. Equation (10) yields the SDR probabilities Pa.2(N)

= ~12(m2)[~ool(Vl)/~11(Oa)]

P1.3(N) =

=

(1 - m 2 ) [ 1 / ( 3 - vl)],

823(m3)[ O~oa(V~)/ton( va)] [ o~2( Vz)/O~z2(V2)]

= ( 3 - m 3 ) [ 1 / ( 3 - va) ] [ ( 2 - v 2 ) / ( 3 - 02) 1 . The values of the SDR probabilities are presented in Table 1 for all permissible values of ( m 2 , m 3 ) . Note that the customers are preferentially routed to branch 3 until branch 3 is full (m 3 = 2) whereupon all customers are routed to branch 2. When branches 2 and 3 are both full ( m 2 = 1, m 3 = 2) then a customer completing service at center 1 will be denied entry to Q(V, ~t") and will be returned immediately to center 1. SDR thus provides a product-form solution for a restricted type of blocking. Note that SDR does not model blocking in its true sense where the highest priority blocked customer requesting entry into Q(V, ~ ) is admitted as soon as queueing space at the requested center in Q(V, ~e') becomes available. Instead, SDR blocking at center 1 implies the busy form of waiting. After being denied access into Q(V, ~/'), the customer is returned to center 1, is queued and is serviced before it can again attempt access into Q(V, ¢/'). The remainder of the analysis assumes that the (7, are negative and that the dtb are positive. Without loss of generality let Ct = - 1 so that

~tb(mb)=d,b--mb,

o~tt(vt)=D,--v,,

~ot-,t(vt) =Dt_a,-Vt.

3. Product-form solution

Consider a MQN with SDR and SIR probabilities as described in Section 2 above. The SDR probabilities are assumed to be chain independent. The chain-dependent analysis is presented in [5].

3.1. Joint probability distribution It is shown in Appendix A that the joint probability distribution P ( N , Jt') is given by M

T

P(N, ~¢) = G-I(N, Jg) I-If,(ni) I~ [[2,-1,(v,)/$2tt(vt)] i=]

t=l

I-] b E.~t -a~/t + 1

Atb(mb),

(16)

A.E. Krzesinski / Multiclass queueingnetworks

132

where the normalizing constant G(N, Jg) ensures that the P(N, ./g) sum to unity. The SDR terms are given by

~-~t_lt(Ut) = OJt_lt(Vt - 1)~t_lt(O t - 1), I2,,(v,) = w , , ( v , - 1)~2,,(v,- 1).

ff~t_lt(0) = 1,

Atb(rnb) =8,b(mb--1) Atb(m b - l ) ,

Atb(0 ) = 1 .

I 2 , ( 0 ) = 1.

The terms f,(ni) are given by J

f~(n~) = [n~!/fli(n~)] 1-I ['y~fJ]/n~j!, j=l where a~(n,)~j is the queue-dependent average service rate of a chain j customer at center i and fl~(ni) = ai(ni)fli(n i - 1), fit(O) = 1, and Yij = ~ij/#ij, where 1/#,j is the average service demand of a chain j customer at center i. The ~j are discussed in the next section.

3.2. Visit counts Let g = (e(2),..., e(B)} denote the index set of the entry centers to all the SDR branches and let 9 = { d(2) . . . . . d(B)} denote the index set of the departure centers from the SDR branches. It is shown in Appendix A that for the entry and departure centers whose indices i are in g + 9 , the ~,j are given by ~ij ~ej where e is the entry center to Q(V, ~ ) . These ~j cannot be interpreted as relative visit counts. For the entry e and the departure center d from Q(V, U ) , ~ej = ~aj. For the remaining centers whose indices i are in . / g - d~- 9 , ~ij = E~kjPkj,~j SO that each of these ~ij can be interpreted as the relative visit counts of chain j customers to center i. =

3.3. Equivalence between state-dependent routing and state-dependent service If each branch consists of a single center then the joint probability distribution (16) reduces to M

T

P(N_,./['[)=G-l(N,.~)Ufi(ni)H[~t-lt(vt)/~tt(vt)] i=l t=l

H

i~ ~/~t--~ + ~

Ati(ni).

(17)

It can be shown [5] that equation (17) is identical to the joint probability distribution for a BCMP network with state independent routing where the set of center indices .At' is partitioned into ( T + 1) disjoint subsets ( . / t ' - ~ , ~e"1 - ~ . . . . . ~e'r-1 - ~/'r, ~c"r) such that the chain j customers at center i in Q(N, . / g ~e'~) have a queue-dependent average service rate ai(ni)l~,j and the chain j customers at center i in Q(N, ~ - ~e~,+l) have a service rate that depends both upon the queue length n, at center i and upon the total population vt within the subnetwork Q(V, ~¢/~t),namely

[a,(n,)/St,(n , - l)][w,t(v t - 1)/Wt_lt(V t - 1)] ~,;.

4. Solution of state-dependent routing networks MQNs with product-form SDR as described in Section 3 above are solved by a combination of convolution and Mean Value Analysis (MVA) methods. The following section presents a summary of the solution method for product-form SDR networks where each branch consists of a single BCMP center. The general case where each branch consists of several BCMP centers is presented in [6].

4.1. MVA solution of Q ( N -

V, . / g - Y/')

The centers in Q(N, J g - ~') are not subject to SDR. Standard MVA and convolution methods are applied to compute the network performance measures such as the chain j queue lengths Q i j ( N - V,

A.E. Krzesinski / Multiclass queueingnetworks

133

J r - ~/') at each center i where i ~ J t ' - ~¢:, the chain j throughputs T j ( N - V, ~/4- ~e-) and the network normalising constants g( N - V, J r - ~').

4.2. Solution of Q(V, "//') The centers in Q(V, ~e') are subject to SDR and a modified version of MVA is applied. Let ~r[Q(N, ~t') : i ~ ~e'] denote a set whose members are the average performance measures for those centers i in the network Q(N, J4) where i ~ ~/'c~¢. The following algorithm summarises the computation which yields the required performance measures ~r[Q(V, ~ ' ) : i ~ "//']. fort=T,

T-1

. . . . . 1:

A: Solve the subnetwork Q(V, ~ - ~ + a )

in isolation to obtain the performance measures ~r[Q(V,

~t -- ~t+l) : i E ~t -- ~t+l]" B: Convolute the performance measures

~r[Q(V, ~ - ~tt+l) : i ~ ~t - Y~t+l] to obtain the performance

measures ~r[Q(V, ~ ) : i ~ ~ - ~+1]C: Convolute the performance measures ~r[Q(V, ~t+l) : i ~ ~t+a] to obtain the performance measures 7r[Q(V, YPtt): i ~ ~t+l]. Then

~r[Q(V, ~t ): i 6 ~t ] = ~ r [ Q ( V , ~t ) : i E Y~t+l] U~r[Q(V, ~t ) : i e ~ t - ~'~+,] next t

The remainder of this section presents a detailed description of the solution of Q(V, ze-).

4.2.1. MVA solution of Q(V, Y~t- ~t+l) For each subnetwork index t, 1 ~< t ~< T: For each value of the closed chain population vector V in the range 0 ~< V~< N : The average waiting time of a chain j customer at center i where i in zc~t- YFtt+1 is given by

Wij(V,~t- ~t+l) = [.ij(dti-Qi(V- lj,~t- ~t+l))]-I v × E n[Sti(n--1)/ai(n)]ei(n--l: V - l j , ~ t - ~ t + , ) , n=l

where Qi(V, Y~t- Y~t+a) is the average queue length at center i and Pi(n : N, zc~t- ~t+a) is the queue length distribution at center i. Let T/j( V, Y~t- Y~t+!) denote the throughput of chain j customers at center i where i in YPtt- Y/~,+~- The average chain j queue length at center i is given by

Qij( V, ~ t - ~t+l) = Tij(V, Y~tt- "~t+l)W/j( V, ~ t - ~t+l)

=~ij[dti--Qi(V-lj,~t- ~t+l)]Tj(V, ~t- ~t+l)W/j( V, ~t- ~t+l)Summing over the centers i yields

Tj(V' ~t -- ~t+l) = Vj/

E

~ij[dti-Qi(V-lj,~t-~t+l)]Wij(V,~t-~t+l)

•

li~,,-~+l The queue length distribution at center i where i in Y~t- ~ + a

e,(n: V, ~ t - ~ t + l )

is given by

J = [~ti(n-- 1)//Oil(n)] E "~,jTj(V, ~ t - ~ t + a ) P , ( n - 1 j=l

: V-Ij,

4.2.2. Convolution solution of Q(V, ~t) Initialise: for each value of the closed chain population vector V in the range 0 ~< V~< N: G(V, Y~t ) = [Wv-tr( v - 1 ) / ~ 7 " r ( V - - 1)] G ( V - lj, Y/'r)/Tj(V- l j, ~t"r) for any j = 1 . . . . . J, l j ~< V.

~ t - zeSt+a)

134

A.E. Krzesinski / Multielass queueing networks

Main Loop: For each subnetwork index t = T - 1..... 1: For each value of the closed chain population vector V in the range 0 ~< V ~
~)/j(V, W, ) =

L~V

The unnormalised chain j throughput is given by

Tj ( V, ~t ) = E Tj ( V - L , ]c/t- ~tt+l)P(L, ~t+l : V, ~t ), L~v where the unnormalised probability of there being L customers in Q(L, Wt+a) given V customers in Q(V, ~ ) is given by : v, )= [&_,,(v)/&,(v)]g(v-i., The normalising constant G(V, W,+I) is available from the previous (t + 1)st iteration of Section 4.2.2. The normalising constant g( V - L, YT,- Wt+ 1) is given by

g ( V - L, Y / T t - ~ + , ) = g ( V - L -

lj, Y/~t-~+I)/Tj( V - L , ~ t - Y/7,+1).

The normalised chain j performance measures are now given by

Qij( V, Y/Tt) = Qij(V, ~t )/G(V, YPtt), i ~ ~/tt-- ~t+l,

Tj(V, ~ )= Tj(V, ~ )/G(V, YPtt),

where

G(V, ~t )= E P(L, ~t+l : V, ~tt ). L~V

4.2.3. Convolution solution of Q( V, Y') The throughputs Tj(V, Y') where ~ = ~1 are already computed in Section 4.2.2 above. The queue lengths Q,j(V, U ) where ~ = ~1 and i in ~1 - ~2 are also computed in Section 4.2.2 above. For each subnetwork index t = T - 1, T - 2 ..... 1: For each value of the closed chain population vector V in the range 0 ~< V ~
Qij(V,W,)=G-I(V,W,,) Y'. Q,j(L,W,+I)fi(L,W,+ 1 : V , ~ ) ,

i~Wt+ 1.

L~V

Completion of Section 4.2.3 yields the required average queue lengths Q~j(V, Y/') where Y/'= Y/'I and i ~ Y/'. The throughputs T~j(V, z¢') are given by

Tij(V, V ' ) = g ; i j [ d l i - Q i ( V - l j , ~")]Tj(V, U'),

i~Yf.

4.3. Convolution solution of Q( N, ~t) A final set of convolutions yields the required performance measures for all centers i in Q(N, .~),

Qij(N, .///)= G - I ( N , .//4) Y' Q i j ( N - V, ~gl- Y/')P(V, zCP:N, JI), V~N

Qij(N, .ell)= G - I ( N , .At') Y'. Qij(V, Y/')P(V, Y/-: N, .~tl), i ~ , VEN

Tj(N, J ¢ ' ) = G - I ( N , .//{) Y' T j ( N - V, ...//-..~)fi(V, $-"': N, ..4t'), V~N

where

P(V, "Yr: N, J/l) = g ( N -

V, ..~t'-"Y/')G(V, $/') and

G(N,

N,

E V'~N

i~-Y/',

A.E. Krzesinski / Multiclassqueueingnetworks

135

4.4. Computational complexity and storage requirements Let M be the number of centers in Q(N, ~¢[) and let V = (V 1. . . . . Vj) denote the population bound on Q(V, 3¢'). It can be shown [5] that the number of arithmetic operations required to compute the performance measure of a product form SDR network containing T nested subnetworks is bounded above by JTM(V~V2... Vj) 2. It can also be shown [5] that the storage requirements of the S D R MVA algorithm are bounded above by M ( M + V)VIVz... Vj where V = Va + . . - + Vj. The exact solution of product-form S D R M Q N s thus becomes computationally intractable for large values of the population vector V. However, it can be shown [5] that if T = 1 so that only one level of subnetwork nesting is present, the computational and storage requirements of product form S D R are of the same order as those of standard MVA [81.

5. Performance impact of state-dependent routing This section presents a summary of the performance impact of product form state dependent routing. The properties of S D R are evaluated by comparing the performance measures of two networks, one with S D R and the other with suitably chosen State-Independent Routing (SIR). Apart from the routing probabilities, the two networks are identical.

5.1. SDR with one level of subnetwork nesting Consider a single-class closed queueing network Q(N, .,¢g) consisting of a central server and four peripheral centers where N is the total number of customers and J r ' = {1, 2, 3, 4, 5} is the index set of the centers. Let each center be a fixed rate F C F S exponential server. Let the peripheral centers define four branches each consisting of a single center. Define a single level of subnetwork nesting so that .,¢g- ~e'= {1} and ~e'= ze"a = {2, 3, 4, 5}.

5.1.1. Adaptive routing with balanced loading Let the central server have an average service rate ~1 100 and let the peripheral centers i = 2, 3, 4, 5 have identical average service rate #i =/~p. Let the routing coefficients dl~ be given the value d~ = d for 2 ~< i ~< 5. Equation (10) yields the SDR probabilities =

Pl.i(N) = ( d - n i ) / ( 4 d - V),

2 <~i <~5,

where n i denotes the population at center i and V = v 1 = n 2 + n 3 + n 4 + n 5 denotes the population in the subnetwork Q(V, ~ ) . A corresponding set of SIR probabilities is obtained from equation (15), 1 Pl.i = a,

2
Note that the S D R probabilities constrain each peripheral queue to contain maximally d customers. In addition, the SIR probabilities route the customers uniformly amongst the peripheral centers - hence the term balanced loading. Let Tso R and Tsm denote the throughput at the central server when customer routing is performed using SDR and SIR respectively. The throughput gain R = 100(Tso R - Tsm)/Tsm defines the percentage increase in throughput obtained when using S D R rather than SIR. Fig. 4 plots the throughput gain R as a function of I*l/t*v for several values of the network population N [10]. The value of the routing coefficient is kept fixed at d = 1. When ~p >>/~1, the peripheral queue lengths ni, 2 ~< i ~< 5, are negligible so that P~.i(N) - PLy- Fig. 4 confirms that S D R and SIR both achieve the same system throughput as #l/~tp---> O. An increase in the ratio i~1/1~p causes the peripheral queue

A.E. Krzesinski / Multiclass queueing networks

136 70

dli=(1,1,1,1)

N=4

80 Z 50

N=3 40 o o Izl

30

M U

20

N*2 Jl

I

z

I

I

?

I

I

I

8

4

I

I

z

12

CPU/IO SP]81¢D IUkTIO Fig. 4.

lengths to increase. However, the SDR probabilities route customers to the empty peripheral queues while allowing each peripheral queue to contain maximally one customer. The SIR probabilities on the other hand do not prevent the routing of customers to already congested peripheral queues. The throughput gain R therefore increases as the population N and as the ratio/~l//~p increase. Figure 5 plots the throughput gain R as a function of/xl//~ p for several values of the routing coefficient d. The queueing network remains as defined above and the network population is fixed at N = 4. Fig. 5 reveals that as the ratio/~l//~p increases, thereby causing the peripheral queue lengths to increase, the SDR probabilities achieve a greater throughput than SIR. Note that SDR becomes less effective as d increases. In the limit d >> n i the SDR probabilities reduce to the SIR form.

70 N=4 60 Z

50 ilu II1

4,0

o

~

ao

N

2O

M

eu 10

4

8 C P U / I O SPEED RATIO Fig. 5.

1:8

A.E. Krzesinski / Multiclass queueing networks

137

5.1.2. Adaptive routing with biased loading Let the central server have an average service rate/*a = 100 and let the peripheral centers have dissimilar average service rates/*i = 20, 40, 40, 20 for i = 2, 3, 4, 5 respectively. Let the routing coefficients be given the values dli = 1, d, d, 1 for i = 2, 3, 4, 5 respectively where 1 ~< d ~< 10. Equation (10) yields the SDR probabilities ( 1 - n , ) / ( 2 d + 2 - V),

i = 2, 5,

( d - n , ) / ( 2 d + 2 - V),

i=3,4.

PI'i(N)=

A corresponding set of SIR probabilities is obtained from equation (15), /1/(2d+2),

i=2,5, i=3,4.

Pl,i= ~ d/(2d + 2),

Note that the SDR probabilities constrain queues 2 and 5 to each contain maximally 1 customer, and queues 3 and 4 to each contain maximally d customers. The SIR probabilities also preferentially route customers to centers 3 and 4 - hence the term biased loading. Fig. 6 reveals that under heavy load conditions ( N = 4) as d increases, both the SDR and the SIR probabilities preferentially route customers to the faster centers 3 and 4. The throughput ratio T3/T2 increases linearly (almost linearly for SDR) as d increases and each set of routing coefficients (1, d, d, 1) results in similar values for Ta/T2 for both SDR and SIR. Fig. 7 plots the throughput T 1 = T2 + T3 + T 4 + Ts (where T/is the throughput at center i) as a function of the one-level routing coefficients. Fig. 7 reveals that both for light ( N = 2) and for heavy ( N = 4) load conditions SDR achieves greater throughput than SIR. However, as d increases, the SDR probabilities preferentially route customers to the faster centers 3 and 4, queues 2 and 5 being constrained to maximally one customer each. The resultant congestion at queues 3 and 4 progressively diminishes the throughput advantage of SDR over SIR. Note that for N = 4 the system throughput under SIR improves and then deteriorates as more

1! 10

R=4

9

\ 8

o 7 6

5 0

4 3 2 1

,

2

4

6

ROUTING COEFFICIENTS Fig. 6.

8

10

A.E. Krzesinski / Multiclass queueing networks

138 75

Ui=(20,40,40,20)

70 65 N=4

I= o

60

0

J

o I=

- - ~

O

G

&

&

A

A

.~----

U

.17

U

I

i

SDR (1,d,d,l)

55

=1 50

SIR ( 1 , d , d , 1 )

--V--

0

m ~3

45

J 40

35

N=2

J

S i

y

i

2

i

i

4

I

i

8

6

10

ROUTING COEFFICIENTS d Fig. 7.

customers are routed to the faster centers 3 and 4. In contrast, for N = 4 S D R p e r f o r m s best when the traffic is equally balanced (d = 1) over the peripheral centers.

5.2. SDR with two levels of subnetwork nesting Consider a single-class closed queueing network consisting of a central server and four peripheral centers. Let each center be a fixed rate F C F S exponential server. T h e peripheral centers define four branches each consisting of a single center. Define two levels of subnetwork nesting where ~e"1 = (2, 3, 4, 5 ) and ~ = (3, 4). The network population is fixed at N = 4. Let the central server have an average service rate #1 = 100 and let the peripheral centers be assigned dissimilar average service rates/~i = 20, 30, 30, 20 for i = 2, 3, 4, 5 respectively.

5.2.1. Adaptive routing with balanced loading Consider the two-level routing coefficients (1, 1, 1, 1) (d, d ) which notation is used to denote dli = 1 for 2 ~< i ~< 5 and d2i = d for i --- 3, 4. Equation (10) yields the S D R probabilities Pl,i( N ) =

( (1 -- n i ) / ( 4 -- Vl), [(d-ni)/(a-va) ][(2-vE)/(2d-v2)],

i = 2, 5, i=3,4,

(18)

where v I = n 2 + n 3 + n 4 + n 5 and v2 = n 3 + n 4. Figs. 8 and 9 plot the throughput ratio T3/T2 and the throughput T 1 = T2 = T3 + T4 + T5 (where T, is the throughput at center i) as a function of the two-level routing coefficients. F r o m equation (18), the S D R probabilities constrain queues 2 and 5 to each contain maximally 1 customer, and queues 3 and 4 to each contain maximally d customers. However, queues 3 and 4 together m a y contain maximally two customers irrespective of the value of d. If u 2 = 2 then no customers are routed to centers 3 and 4. In addition, if 02 = 2, then P1,2(N)

-I- P1,5 ( N )

= (1 - n 2 ) / ( 4 -/31) q- (1 - n 5 ) / ( 4 - u1) = ( 2 -- n 2 - n 5 ) / ( 4 - vl) -- 1,

so that all customers are routed to centers 2 and 5. Fig. 8 confirms that two-level S D R with routing coefficients (1, 1, 1, 1) (d, d ) does not preferentially route customers to centers 3 and 4. Fig. 9 reveals that two-level S D R achieves a greater throughput than one-level SDR.

139

A.E. Krzesinski / Multiclass queueing networks 10

N=4

9

ui=(20,30,30,20)

8

0

7 6

~4

5 0

4 3 2

[]

1

i

[] i

2

[]

[]

I

4

i

0 i

[] !

SDR 1,1,1,1)(d,d) [] 0

8 ROUTING COEFFICIENTSd Fig. 8.

f

8

10

5.2.2. Adaptive routing with biased loading

Consider the two-level routing coefficients (1, d, d, 1) (1, 1). Equation (10) yields the SDR probabilities Pld( N )

= f [(1 - n i ) / ( 2 d + 2 - ua) ], [(1

n~)/(2d+2

i = 2, 5,

v1)][(2d-v2)/(2-v2)],

(19)

i=3,4.

F r o m e q u a t i o n (19), the S D R p r o b a b i l i t i e s c o n s t r a i n e a c h c e n t e r i, 2 ~< i ~ 5, to c o n t a i n m a x i m a l l y o n e c u s t o m e r . If v 2 = 2, t h e n n o c u s t o m e r s are r o u t e d to c e n t r e s 3 a n d 4. I n a d d i t i o n , if v 2 = 2, t h e n

P1,2(N) + P1,5 ( N ) = (1 - nz)/(2d+ 2 - vl) + (1 - ns)/(2d+ 2 - Va) = ( 2 - n 2 - n s ) / ( 2 d + 2 - Va) = ( 2 - n 2 - n s ) / ( 2 d - n 2 - ns) = F < I. 89 88

8'7

~

N=4~

ui:(2°'3°'3°'2°)

68 65 64

SD°R( I"1'ID I) (d'd)D

63

0

6261-

6059 58 ° 575fl55 54 53 5Z 51 50 49 2

4

8

ROUTING C O E ~ C I E H ' I ' S d

Fig. 9.

8

10

140

A.E. Krzesinski / Multiclass queueing networks 120 110 lO0

gO 80 70

R: 0

80

0 n

50

[.,

4O &

SDR CPU THROUGHPUT

30

SDR PERIPHE'AL THROUGHPUT

,°//

2O

0 0.0

i 1.0

SIR THROUGHPUT SIR THROUGHPUT WITH BLOCKING

I

I

I

I

I

I

I

I

:~.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

MULTIPROGRAMMINC LEVEL Fig. 10.

Therefore, in contrast to the balanced S D R probabilities presented in equation (18) above, when centers 3 and 4 are full the biased S D R probabilities presented in equation (19) route only a fraction F of the customers to centers 2 and 5, the remainder (1 - F ) being returned immediately to center 1 where after completing service they can again request entry to the centers in Q(V, ~/'). Fig. 8 confirms that as d increases, two-level S D R with routing coefficients (1, d, d, 1) (1, 1) preferentially routes customers to centers 3 and 4. Figs. 8 and 9 reveal that, in comparison with the one-level routing coefficients, the two-level coefficients sustain a higher throughput while routing fewer customers to the faster centers 3 and 4.

5.3. SDR and blocking Consider a single-class queueing network as in Section 5.1 above consisting of a central server and four peripheral servers. Let each center be a fixed rate FCFS exponential server. Let the central server have an average service rate #a = 100 and let the peripheral centers have identical average service rates/x i = 20 for 2 ~< i ~< 5. Let the peripheral centers have identical average service rates /~i = 20 for 2 ~< i ~< 5. The peripheral centers define four branches each consisting of a single center. Define a single level of subnetwork nesting where ~"= (2, 3, 4, 5} and assign the routing coefficients the values dl~ = 1 for 2 ~< i ~< 5. Equation (10) yields the S D R probabilities PI,i(N) = (1 - ni)/(4 - V), 2 ~< i ~< 5, where V = n2 + n3 + n4 + n s .

Fig. 10 plots the SDR CPU throughput T 1 and the peripheral S D R throughput T = T2 + T3 + T4 + T5 (where Tj is the throughput at center i) as a function of the multiprogramming level N. Fig. 10 emphasises a significant feature of SDR, namely that customers which are denied entry into the S D R subnetwork Q(V, ~ ) are not queued for admission but are (in this example) returned to the central server where they complete another service interval before again attempting access into Q(V, Y/). Thus, for N > 4 there are always one or more customers at the central server and hence T 1 = 100. Fig. 10 also presents the SIR throughput for a product form queueing network with Pl,g = ~, as well as the SIR throughput for a SIR queueing network where PI,~ = -14and in addition a concurrency constraint is placed on the subnetwork Q(V, :r) so that maximally four customers can be concurrently present in

A.E. Krzesinski / Multiclassqueueingnetworks

141

Q(V, y/). Once v = 4, customers are queued for admission in F C F S order to Q(V, ~ ) . Queueing networks with concurrency constraints violate product form, and approximate solution methods have been developed [4] for the accurate and efficient solution of such networks. Fig. 10 reveals that S D R achieves greater throughput than SIR. The blocking imposed by S D R where (in this example) each peripheral center can contain maximally one customer results in a greater throughput than that achieved by concurrency constrained SIR which, although it also allows maximally four customers into the constrained subnetwork, unlike S D R cannot prevent the routing of customers to already congested peripheral queues. As mentioned above, the S D R peripheral throughput rate T = T2 + T3 + T4 + T5 is constant for N >/4. The SDR peripheral throughput rate will be constant only if the central server has a fixed (load independent) average service rate. This can be demonstrated as follows. Let center 1 have a fixed average service rate #1 and let nl denote the number of customers at center 1. If N = 4 then 0 ~< n 1 ~< 4 and the S D R subnetwork is subject to a Poisson arrival process )~(nl) where ~ ( n l ) = 100 for 1 ~< n 1 ~< 4 and 2~(0) = 0. Consider N = 5. Maximally four customers can be in Q(V, Y/') so that 1 ~< n I ~< 5. The SDR network is now subject to a Poisson arrival process with parameter ) k ( n I - - 1 ) identical to the Poisson process described above with population N = 4. The S D R peripheral throughput is thus constant for N >/4, provided that center 1 has a fixed average service rate.

6. Conclusion The paper presents product-form solutions for multiclass queueing networks with a certain type of state-dependent routing. The requirement that the M Q N has a product-form solution determines that the centres in the M Q N are partitioned into disjoint groups called branches and that the branches are further arranged into a hierarchy of nested subnetworks. The S D R probabilities are shown to be products of linear functions of the branch and subnetwork populations. A consequence of SDR, namely the imposition of population bounds upon the branches and upon the subnetworks is examined. The relationship between S D R and a form of blocking is investigated. The paper proceeds to solve the global balance equations and to derive the joint probability distribution for a M Q N with product-form SDR. MVA methods are then presented for computing the performance measures for a M Q N with product-form SDR. The s p a c e - t i m e requirements of the S D R MVA algorithm limit exact solutions to small multichain populations. The paper concludes by presenting a numerical investigation of the properties of product-form SDR. Several examples are used to examine the performance advantages of product form S D R over state-independent routing.

Appendix A. Global balance equations This appendix presents the global balance equations for a central server M Q N with chain-independent SDR. The global balance equations are solved to yield a product-form expression for the joint probability distribution. Let Q(N, ..¢[) denote a closed central server M Q N consisting of M F C F S service centers with index set J [ = {1 . . . . . M ) . Let the customers belong to J closed chains labelled (1 . . . . . J ) . Let N = (N~ ..... Nj) denote the population vector where Nj is the number of chain j customers. Let nij denote the number of chain j customers at center i and let n i = (nil + • • • + n i j ) denote the total n u m b e r of customers at center i. The network state descriptor is given by N = (n 1. . . . . nM) where the population vector at center i is given by n i = ( n i l . . . . . n i j ). Let l j denote a unit vector in the j direction. The set of center indices ..¢t' is partitioned into two subsets ~ t ' - ~e" and Y/" where ~ ' = (2 . . . . . M - 1). The subnetwork Q(V, ~e') contains all the centers which are subject to SDR. Let the central server center 1 be the entry center to Q(V, ~e-) and let center M be the departure center from Q(V, Y/').

142

A.E. Krzesinski / Multiclass queueing networks

Define a sequence of sets ( Y"I. . . . . Y:r } of center indices such that ff = Y/r+ 1 c Y/'r c Y/r- 1 c .. • c Y/'I = Y/'. The sets ~ are chosen such that for any center index i, 2 ~< i ~< M - 1, there exists a unique subnetwork index t, 1 ~< t ~< T, such that i in ~t - Y/~t÷l. Let PI.~(N) denote the SDR probability of a customer, upon completing service at center 1 being routed to center i where 2 ,%
[ PI,i(N) =

' 8ti(l'li)Hs=1%s(v,) if%~(Vs)>Of°ralll~
t0

if %~(G) = 0 for any 1 ~
Let P(N, J r ' ) = P ( t i 1 . . . h i . . . tiM, .A[) denote the joint probability distribution. The global balance equations are given by M

P(N,

E

=

i=l M-1 J

= E

EP(til+ls."ti~-lj..'nM,"g)a~(nl+l)~l[(naj+l)/(nl+l)]

i=2 j = l

1)

ls ( Vs 1) s=l lZI 60s~'~s( ~s -----i )

x

(A.1)

J

+ ~.,P(n,+lj...nM--lj,..4()al(nl+l)~l[(nlj+l)/(nl+l)] j=l

× 1M-1

~ti(ni) __

OOs-ls ( Us)

(A.2)

J

+ ~_, ~ _ P ( n a . . . n , + l j . . . n M - - l j , . / g ) a i ( n i + l ) l x i ( n i j + l ) / ( n i + l i=2 j = l J + E P(ti1j=l

lj'''tiM+

lj, ~)OIM(nM+

1)~M(nMj+ 1)~(riM+ 1),

)

(A.3)

(A.4)

where the terms (A.1), (A.2), (A.3), and (A.4) describe the following transitions: (A.1): transition of a chain j customer from center 1 to center i, (A.2): transition of a chain j customer from center I to center M, (A.3): transition of a chain j customer from center i to center M, (A.4): transition of a chain j customer from center M to center 1. Four successive changes of variable are required to solve the global balance equations and yield a product-form expression for the joint probability distribution P ( N , ,/g). Let M P ( N , Jg) = I-I [ 1 / f l ( n , ) ] Q ( N , ,ff~), where fli(ni) =ai(ni)fl~(n ~- 1) and fl~(0) = 1. i=1 This change of variables allows a common factor I-[i=lfli(ni) M to be removed from the global balance equations. Let T Q(N, .../t')=I-[ 1-I At,(n,)R(N , .AI), t=l i~ ~t-- ~t+ 1

w h e r e A . ( n i ) = 8 . ( n i - 1) Ati(n i - 1) and Ati(O ) = 1.

A.E. Krzesinski / Multiclass queueing networks

This change of variables allows a common factor 1-ltr=lFI,~ G _ ~ , + , balance equations. Let

143

Ati(ni) to be removed from the global

T

[~t-lt(Ut)/~tt(V,)] S(N, ,/~),

R(N, ,AI)= H

t=l

where ~'~t_lt(Ot) = 03t_lt(U , -- 1)~2t_,t(v t - 1)

and

~"~t_lt(0) =

1,

12,t(ot) = o : , ( v , - 1)12tt(v t -

1) and 12,t(0 ) = 1. This change of variables allows a common factor FIT=1[ ~2t-l,(vt)/12, (vt)] to be removed from the global balance equations. Let S(N,

M J/~¢)= Hf/(n,), i=1

J where f/(n/)=n!

I-I ( ~ i j / # i ) n i J / n ! i j • j=l

This change of variables allows a common factor I-l~lf.(n,) to be removed from the global balance equations. After applying the four changes of variables the global balance equations become

M

J

E

= E

,=1

OLi(l~i)(~lj/~ij)]£'

j=l

,

]

1 - E 8,(.,)

+ E J=l

i=2

M-1 J

s=l

~,,I, Us)

fl

+ E E CtM(nM)(~ij/~Mj)IXMS,(n,) i=2j=1

O:'--ls(V')

s=l

03ss(Us)

J

+ E

j=l

which have the s o l u t i o n ~lj ~- ~Mj and ~ij = ~lj for 2 ~< i ~
P(N, Jr) = G-I(N, ~ )

T

Hf,(FI,) H

i=l

t=l

[~'~t--lt(Ut)/~'~tt(Ut)]

H

A,,(.,),

where the normalising constant G(N, Jr) ensures that the P(N. Jr') sum to unity. References [1] E. Baskett, K. Chandy, R. Muntz and P. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. ACM 22 (2) (1975) 248-260. [2] J. Buzen, Computational algorithms for closed queueing networks with exponential servers, Comm. ACM 16 (9) (1973) 527-531. [3] K. Chandy and D. Neuse, Linearizer: A heuristic algorithm for queueing network models of computer systems, Comm. A C M 25 (2) (1982) 126-134. [4] E.D. Lazowska, J. Zahorjan, G.S. Graham and K.C. Sevcik, Quantitative System Performance (Prentice-Hall, Englewood Cliffs, N J, 1984). [5] A. Krzesinski, Multiclass Queueing Networks with State Dependent Routing, IBM Res. Rept. RC-9761, Yorktown Heights, NY, December 1982.

[6] A. Krzesinski and P. Teunissen, An Approximate Solution Method for Multiclass Queueing Networks with Adaptive Routing, Rept. ITR 85-04-00, Inst. for Applied Computer Science Univ. of Stellenbosch, South Africa, 1985. [7] M. Reiser and H. Kobayashi, Queueing networks with multiple closed chains: Theory and computational algorithms, IBM J. Res. Develop. 19 (3) (1975). [8] M. Reiser and S. Lavenberg, Mean Value Analysis of Closed Multichain Queueing Networks, IBM Res. Rept. RC-7023, Yorktown Heights, NY, March 1978; also in: J. ACM 27 (2) (1980) 313-322. [9] M. Reiser, Mean value analysis and convolution methods for queue dependent servers in closed queueing networks, Performance Evaluation 1 (1) (1981) 7-18. [10] D. Towsley, Queueing network models with state-dependent routing, J. A C M 27 (2) (1980) 323-337.