Mean-Value Analysis of Closed Multichain
Queuing Networks
M. REISER
IBM Zurich Research Laboratory, Ruschhkon, Switzerland
AND
S. S. LAVENBERG
IBM Thomas J. Watson Research Center, Yorktown Heights, New York
ABSTRACT. It tS shown that mean queue sizes, mean waiting tunes, and throughputs in closed multiple-cham
queuing networks which have product-form solution can be computed recurslvely without computing product
terms and normahzatton constants The resulting computational procedures have improved properties (avoidance
of numerical problems and, m some cases, fewer operations) compared to previous algorithms. Furthermore, the
new algorithms have a phystcaUy meaningful interpretation which provides the basis for heuristic extensions that
allow the approximate solution of networks with a very large number of closed chains, and which is shown to be
asymptottcaUy vahd for large chain populations.
KEYWORDSAND PHRASES. queuing networks, closed queuing systems, analytical models, performance evaluation, algorithms for the solution of queuing networks, approxtmaUon for general queuing networks
CRCATEGORIES: 4 6, 5 5
1. Introduction
T h e traditional a p p r o a c h to the solution o f M a r k o v i a n q u e u i n g networks was to f o r m u l a t e
a system o f algebraic equations (balance equations) for the j o i n t probability distribution o f
the vector-valued system state. It was the important discovery o f Jackson [7], G o r d o n and
N e w e l l [6], and Baskett, Chandy, Muntz, and Palacios [2] that for certain classes o f
networks, the solution o f the balance equations is in the f o r m o f a product o f simple terms.
All that r e m a i n e d to be d o n e numerically was to normalize the product terms to form a
proper probability distribution. In the case o f networks with closed routing chains, however,
this normalization turned out to be computationally limited, that is, until the publication
o f the convolution algorithm by Buzen [4] and Reiser and K o b a y a s h i [1 1, 13, 14].
F o r practical purposes, however, the j o i n t distribution contains far too m u c h detail.
M u c h simpler quantities such as m e a n q u e u e sizes, m e a n i n g waiting times, utilizations,
and throughputs were needed. W i t h i n the f r a m e w o r k o f the convolution algorithm [4, 1 1,
13, 14] it has been shown that such quantities can be derived f r o m the n o r m a l i z a t i o n
constants in an efficient m a n n e r (in m a n y cases this requires an effort similar to the
calculation o f the normalization constant). T h e m e a n - v a l u e algorithm given in this p a p e r
works directly with the desired statistics. Its operations count is asymptotically e q u i v a l e n t
to earlier algorithms. H o w e v e r its p r o g r a m i m p l e m e n t a t i o n is often simpler.
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Authors' addresses' M Reiser, IBM Zunch Research Laboratory, CH-8803 Ruschhkon, Saumerstrasse 4,
Switzerland, S S. Lavenberg, 1BM Thomas J. Watson Research Center, Yorktown Heights, NY 10598
© 1980 ACM 0004-541 !/80/0400-0313 $00 75
JournaloftheAssoclaUonforComputingMachinery,Vol 27,No 2, April1980,pp 313-322
314
M. REISER AND S. S. LAVENBERG
The mean-value analysis is based on a relation between the mean waiting time and the
mean queue size of a system with one customer less. This mean-value equation, augmented
by Little's equation applied to each routing chain and separately to each service center,
will furnish a set of equations which are easily solved numerically. At no point do we
compute normalization constants. The new algorithms are simple and avoid overflow/
underflow problems which may arise with the convolution algorithm. All mean values for
the queue size are calculated in parallel. Hence the storage requirement is higher than with
previous algorithms. However, the new method is considerably faster in the case o f
multiservers.
Furthermore, our algorithms have a physically meaningful interpretation which provides
the basis for heuristic extensions. We give one such heuristic extension that replaces the
recursion over the closed chain populations by an iteration for fixed populations and thus
makes possible the approximate solution of large problems with possibly hundreds of
closed chains. Such problems arise in the framework of communications network modeling
and are intractable by exact methods. The heuristic is shown to be asymptotically valid for
large chain populations.
2. Preliminaries
We consider closed multichain queuing networks which have a product-form solution. In
particular, we assume R closed routing chains and L service centers. Each chain contains
a fixed number of customers who proceed through a subset of the service centers according
to a Markov chain. Service centers adopt one of the following service mechanisms:
(i) FCFS: customers are served in order of arrival; multiple servers are allowed.
(ii) PS: customers are served in parallel by a single server, each customer receiving an
equal share of the service rate (processor sharing).
(iii) LCFSPR: customers are served in reverse order of arrival by a single server and the
last arrival preempts the customer in service who will later resume service at the
point of interruption (last-come first-served preemptive resume).
(iv) D: customers are delayed independently of other customers at the service center
(there is effectively an infinite number of servers).
In the case of an FCFS service center, service demands must be exponentially distributed
and independent of a customer's chain membership. All other types of service center admit
chain-dependent phase-type service demand distributions [5, 9]. The service rate of a
service center may depend on the number o f customers at the center. One way this can
occur is if there are multiple servers at an FCFS service center, each having a constant
service rate. We adopt the following notation:
gr:
population size of chain r
K ffi (K1, K2, . . . , KR): population vector
Tr, l: mean value of the service demand of a customer in chain r at
service center 1
re(k): service rate of service center l as a function of the number of
customers at the center
1"(0: an arbitrarily chosen service center visited by chain r
Or,t: mean number of visits a customer in chain r makes to service
center 1 between successive arrivals at service center l*(r)
S(r): set of service centers visited by chain r
R(i): set of chains which visit service center 1
~r,l: throughput of chain r customers through service center !
P/ routing matrix of chain r
nr, l: equilibrium mean number of chain r customers at servme center l
hi: equilibrium mean queue size of service center I
Mean-Value Analysis of Closed Muitichain Queuing Networks
315
Wr,t: equilibrium mean wmtlng time (including service time) o f a chain
r customer at service center 1
Mr: number o f servers at service center 1
ut: equilibrium mean number o f busy servers at service center 1
(utilization in the case of a single server)
pt(k): equilibrium marginal queue size probability o f service center l
(probability k customers at center)
e~: R-dimensional umt vector in direction r
When convenient, we shall use K as an argument to indicate that the quantities relate to
the system with population vector K, e.g., nt(K) for a mean queue size or pz(k, K) for a
marginal queue size probability.
Without loss o f generality, we may assume Pr, the routing matrix o f chain r, to be
irreducible over S(r). F r o m standard Markov chain analysis, the mean number o f visits a
customer makes to service center l between successive arrivals at service center l*(r)
satisfies the equations
Or = OrP,,
(2.1)
Or,t*t~) = 1,
(2.2)
where & is the vector (8~,t: 1 E S(r)). (If ! E S(r), then &,t = 0.) Since Pr is an irreducible
stochastic matrix, the solution to (2.1) and (2.2) is unique. It is known that 8r,t is also the
throughput of chain r through service center l in units of the chain r throughput at the
marked center l*(r); namely,
(2.3)
~r,1 = Or, l~r,l*tr).
It will also prove useful to define traffic intensities
(2.4)
pr,l = Or,l~'r,l.
We shall also use Q(K) to denote a queuing network in the above class with population
vector K.
In order to prove the following theorems, we shall need the product-form solution o f
Q(K), which is well known from [2]. (We assume that the product-form solution is the
unique solution to the balance equations, i.e., that the network evolves as a finite state
Markov process which has a single irreducible class o f states.) Let (kl, ks . . . . . kL) be a state
vector of the network Q(K), where kt = (kt,t, k2,1 . . . . . ks,z) is a state vector o f service center
I and k~ + k2 + . . . + kL = K; kr,t is the number of chain r customers in center I. (The state
vector is not Markovian.) The equilibrium probability o f being in state (k~, k2. . . . . kL) is
given by
Irl(ka)~r2(k2) • • • lrL(kL)/g(K),
(2.5)
where
~'~)
k] k 2
ffi a~(I k 0C(k)p~,~2,~
...
kR
PR,t,
(2.6)
Ikl f kl + k2 + . . . + kR,
(2.7)
a~(O = II ( l / ~ ( j ) ) ,
(2.8)
C(k) = I k I!/k,!k2! . . . ks!,
(2.9)
.I--1
and g(K) is the normalization constant. (We assume 0° = 1.) In all that follows any
quantity (e.g., Irj(k)) whose argument (e.g., k) has one or more negative components is
equal to zero.
There are many recursive relations which relate quantities o f Q(K) to those of
316
M. REISER AND S. S. LAVENBERG
Q(K - e,), i.e., to a system wRh one customer less in chain r. Such a relation for the
marginal queue size probabilities, not found in earlier publications, is given in
LEMMA
I
I
pt(k, K) -- p ~
R
tel
Xr'tOK)¢r'tpt(k --
1, K - er),
k _> 1.
(2.10)
PROOF. The following is a useful expression for the probability pt(k, K) that service
center I is in state k:
pt(k, K) = ~r,(k)ht(K - k)/g(K),
(2.11)
where the quantity ht(i) is the sum of product terms ~rl(i0 . . . ~rz-l(it-1)~rl+l(it+0 . . . 7rL(iL)
over the domain ((il . . . . . it-l, iz+l. . . . . iL):~j,,t il ffi i}(i and ij, j = 1. . . . . L, are Rdimensional index vectors). The term ht(i) is a normalization constant of a network related
to Q(i) with service center l removed. We also use the well-known equation for the
throughput [14], namely,
~,,t(K) = O,,tg(K - e,)/g(K).
(2.12)
For simplicity of notation we drop the subscript I from pt( , ), a t ( ) , a t ( ) , ht(), ~,,t(),
p,,t, ¢r,t, let k = [k [, and let pk = p~ . . . . p ~ . Then, from (2.11) and (2.6),
p(k, K) = pka(k)C(k)h(K - k)/g(K)
R
--
1)
= E PrPk-e" a(__k___ C(k -- er) hOK
~
_
-- e, -- (k -
#(k)
g(K
eJ g0K
~R p r p ( k - e r , K -
1
~(k) ,.-1
1
-
e,))
g(K
e,)
-
e,)
gOK)
-- er)
g(K)
R
=
E
#(k) r-~
¢,A,(K)p(k
-
e,,
K
(2.13)
- e,),
where we used the identity C(k) -- ~ - 1 C(k - e r ) and the fact that h(K - e, - (k - e,))
ffi h(K - k) to obtain the second equality. Equation (2.12) was used to obtain the last
equality.
Equation (2.13) is similar to (2.10), but for the marginal-state probabilitiesp(k, K) rather
than the marginal queue size probabilitiesp(k, K). The lemma follows by taking appropriate
sums, whose details we omit for conciseness. A result similar to (2.10) is found in
[151. []
We now proceed to the main result.
THEOREM 1. The equilibrium mean waiting time Wr.t(K) o f chain r customers at service
center ! is related to quantities o f the network Q(K - e,) by
wr,/(K)=¢,,t
l+nl(K-er)+
~ i
- 1 pt(i - l , K - e r )
.
(2.14)
Ill
PROOF. For simplicity of notation we drop the subscript 1 and argument K from
n,,l(K), w,,t(K), k,.t(K), and drop the subscript l from a t ( ) , p t ( ) , h t ( ) , pt( , ), Or,l, ¢,,l.
We let K = [ K l, i = I i I, and pl = pil . . . p ~ , where i ffi (il . . . . . tR). From the definition of
n, and from (2.11), (2.6), and (2.12),
n¢=
~
i,C(i)a(Op i h ( K -
=
2
er<i.<K
i)
~
0~_~_K
iC(i
-
e,)
-
orOl_e r h(K
-
er -
(i -
gg(K -- er)
er)) y(K
-
y(K)
er)
Mean- Value Analysis of Closed Multichain Queumg Networks
317
i
.
= er<~.t.<._K
~75p(l -- er, K -- er)~rTr
K
~"
•
~rTr ,-,~ [ p ( j -
l, K - e~) + ( j -
l ) p ( j - 1, K - er)
+(i~+j)-J)P(J-I,K-e,)]"
(2.15)
The theorem follows from (2.15) and Little's equation nr -- ~rWr. El
Theorem l relates the mean waiting time to quantities for a network with one customer
less in the chain in question. The relation simplifies for service centers with servers of
constant rate, as stated in
COROLLARY I.
is FCFS let
Let service center 1 have M~ constant unit rate servers. I f service center l
ql -- equilibrium mean waiting line size (excluding customers in service),
PBl = equilibrium probability that all servers are busy.
I f service center ! is FCFS, then
Wr.l(K) "~" "I'r,l+
gr,1
gl
[qt(K - er) + PBt(K - er)].
(2.16)
I f service center l is FCFS single server, PS, or LCFSPR, so that Mt -- 1, then
w~a(K) = rr, l [ l + nl(K -- er)].
(2.17)
PROOV. For an FCFS service center with Mt servers each with unit service rate,
#t(/) = min(i, Mz).
(2.18)
In this case the rate function is simply the number of busy servers. Using (2.18), letting M
= Mr, and using the simphfied notation from the proof of Theorem 1, (2.14) can be written
as
Wr -~ r r
1 + ~
2
(i -- M ) p ( i ,
K
-
er) + g
l
p(i,
K
-
eO
•
(2.19)
But the first sum in (2.19) is the definition of q~(K - er) and the second sum is the definition
of PBt(K e~), so that (2.16) follows. If Mz = 1, then ~l(0 = 1 and (2.17) follows directly
from (2.14). Equation (2.17) is also found in [3]. []
-
Equations (2.16) and (2.17) are so striking that they call for an explanation. Such an
explanation is provided by the main theorem of [8], which states that a chain r customer,
arriving at service center I in the system Q(K), "sees" the system Q(K - er) in equilibrium.
The meaning of (2.16) now becomes clear. It simply states that the mean time a customer
stays at a service center equals his own mean service time plus the mean backlog upon
arrival. In the case of a PS or LCFSPR service center, [l + nl(K - er)] is the expansion
factor due to congestion by other customers.
Finally, we comment about more general customer class concepts. At first, it might seem
that the closed chain model considered here is less general than the model o f [2] where
customers are allowed to change classes and hence a customer in chain r may arrive at a
service center as a member of different classes (with possibly different mean service
demands). However, it was shown by Reiser and Kobayashi [14] that such a more general
M. R E I S E R A N D
318
S. S. L A V E N B E R G
model can be mapped trivially into a multichain model without class changes. Thus a
solution to QOK) is also a solution to models with class changes.
3. Mean- Value Analysis
In this section we will give a recursive mean-value analysis for Q(K). We assume that all
servers have constant unit service rates. We initially assume that all FCFS service centers
have a single server. Later, we will allow multiple server FCFS service centers. The starting
point is Corollary 1, which relates the mean waiting time of the system Q(K) to the mean
queue size of the system Q ( K - er) with one customer less in chain r, namely,
wr,l = ~r,l[l + nl(K - er)].
(3.1)
(We will omit the argument K when it is clear we are referring to Q(K).) Equation (3.1)
holds for service centers of type FCFS, LCFSPR, and PS. From the definiUon of a service
center of type D, it follows directly that
w~,t = ~r,t.
(3.2)
Sufficient additional equations are found by applying Little's equation separately to each
chain and each service center. Let us look at chain r from the marked service center l*(r).
The average number of customers in the chain is K~ (fixed value). The mean time a
customer spends in chain r between successive visits to l*(r) is easily shown to be the sum
over the service centers o f the terms (mean number o f visits to the service center). (mean
waiting time per visit). Thus, Little's equation gives us
K~
~lEstr} O~.lW~,l '
m
hr
(3.3)
where ~ is the throughput o f chain r at service center l*(r), i.e., ~r = ~r,~'t~}. Little's equation
for each service center yields the relation
n~,l ffi ~,lw~,l = ~O~,lw~,~
(3.4)
Now eqs. (3.1) to (3.4) allow for a recursive evaluation of mean queue sizes, mean waiting
times, and throughputs. The starting point is
n~,t(0) = 0
(3.5)
for all r = 1,2 . . . . . R ; l f f i 1,2 . . . . . L.
We shall state the computational procedure in semiformal algorithmic notation. It will
be beneficial to make the substitutions
w~z-- Or,zWr,t,
(3.6)
nT-- 1 + nt.
(3.7)
By i ffi (/1,/2 . . . . . &) we denote an R-dimensional index vector. Index vectors are appended
only to quantities which must be stored in looping through the index vectors. We assume
that reference to elements with negative indices yields the value zero. Furthermore, let
6(0 = 1 if i > 0 and 6(0) = 0.
A L G O R I T H M I. T o c a l c u l a t e m e a n q u e u e sizes, m e a n w a i t i n g times, a n d t h r o u g h p u t s (single server case)
1. ( I m U a h z a t l o n ) n~(0) * - 1 for all I - 1, 2,
2. ( M a r e loops) f o r h ffi 0, 1,
steps 3 to 5.
,L
, K1; t2 ffi 0, l . . . . .
3 ( C o r o l l a r y 1)
fp~,~6(ip)
W~.l <-" ~ pr.ln l*(| -- er)
f o r all r ffi 1, 2 . . . .
R and
If c e n t e r I is t y p e D
otherwise
I E S(r)
K2; . . . ; ZR = 0, 1,
, KR (il c h a n g e s m o s t r a p i d l y ) p e r f o r m
319
M e a n - Value Analysis o f Closed Muitichain Queuing N e t w o r k s
4 (LRtle's equation for chains)
lr
Et~s(,~ W*r,t
f o r a l l r f f i !,2,
.,R.
5 (LRtle's equatmn for service centers)
n~(i)~l+
foralll=
~ ~rWr*,t
rER(I)
1,2, . , L
The operations count for this algorithm is bounded by 2 R L - R additions and
+ R multiplication/divisions or 4 R L operations per recursive step o f which
there are K1K2 . . . K s . This is the same as the convolution algorithm in its most efficient
form [11]. However, Algorithm 1 completely avoids a genuine problem of the convolution
algorithm, namely, that the floating point range of many computers may be easily
exceeded. Scaling, as discussed in [11], may partially alleviate the problem. Yet the scaling
algorithm is complex and does not always work. The authors have seen several well-posed
modeling problems involving relatively large populations (e.g., >100) and type D service centers which were not solvable in the range of floating point numbers 1E ± 75,
despite scaling. The storage requirement is of the order LK1K2 . . . KR as compared to
2KiK2 . . . KR for the convolution algorithm.
We now proceed to extend the computational procedure to handle FCFS service centers
with multiple constant unit rate servers. The mean value equation (2.14) for such a center
can be written
2RL
w,.~ = ~
1 + n , ( g - e,) +
E ( M i -- I -- O?~(i, g - e,)
g~o
,
(3.8)
where et is the mean service demand, which is assumed independent of the chain. The
calculation is complicated by the marginal queue size probabilities, which we have to carry
along in the recurswe scheme. This can be done by means of Lemma 1, which allows
calculation ofpt(i, K), i = 1, 2 . . . . . M~ - 1 from previously computed values. In order to
keep the recursion going, we need an independent equation forpt(0, K), which is obtained
from the relations
Mi
(Mr - Opt(i, K) -- mean number of idle servers = M~ - ut,
(3.9)
where
R
ut = ~'t ~ ~,,t.
(3.10)
r~l
Note that (3.10) is simply Little's equation applied to the set of servers.
We summarize in algorithmic notation:
A L G O R I T H M 2 To calculate mean queue sizes, mean waiting tunes and throughputs (multiple server case):
1 (Intttahzation)n~(O)~--l,pt(O,O)*-l,pt(i,O).~-Oforlffi ! , 2 , . , L , lffi 1,2 . . . . M ~ - l
2 (Mare loops on i) same as Algorithm 1
3. (Corollary I)
,
W,.~
~.= 0 r d
Ml
,.
nl0--e,)+
2 (Ml-l-j)pt(j,i-e,)
j-o
for • ffi 1, 2,
, R and each multtserver FCFS service center ! E S(•); for other service centers use the waiting
tune equations m step 3 of Algonthm 1.
4 (Ltttle's equation for chains yields h,) same as A l g o m h m i
5 (Little's equation for service centers yields n7 (i)) same as Algorithm i
6. (Addmonal step under mare loops to calculate margmal queue stze probabdmes) for each multiserver FCFS
servtce center I andj = 1, 2,
, Mt - 1
320
M. REISER A N D S. S. L A V E N B E R G
I
p ~ , i) ,,---
Y
X , a , . t p ~ - 1, i - e,.).
J r~R(/)
rER{/)
p~0,i)~t-
u,+ 2 (M,-j)p,~j,i).
j--I
The operations count per multiserver service center and per recursive step is of the order
of 2(M + I)R additions and 3 M R + 2 M multiplications or roughly 5 M R operations. We
observe that it grows linearly with M, which is now substantially better than previous
algorithms [l l] which had an operations count of the order M R. Once more we have
avoided normalization constants and their associated growth problems. Storage requirement is of the order MLKiK2 . . . KR, In the case of the convolution algorithm, the storage
requirement is 2KiK2 . . . KR + M R.
Finally, we wish to mention that Algorithm 2 generalizes straightforwardly to general
queue-dependent service rates if (2.14) is substituted in step 3.
4. Heuristic Extensions and Conclusion
A major limiting factor m the use of closed chains is that the computational complexity is
proportional to KiK2 . . . KR. Unfortunately, many computer modeling applications do
require many chains. Examples are central server models with many workload classes (as
subsystems) [1] and communication network models where each session or virtual channel
is mapped into a closed chain representation [ 12], Especially in the case of communication
networks, there may well be hundreds of chains and service centers, which makes them
intractable by the recursive techniques of Section 3 (or by earlier convolution algorithms).
In this section we propose an iterative algorithm, based on Algorithm I, which yields
approximate resuhs,
We introduce correction terms
m.t(K - ej) = nr.z(K) - ¢{-.t(K).
(4.1)
Assume that we had a procedure to calculate ~s(K) from the mean values in Q(K). With
such a function, the recursion (3.1) to (3.5) converts into a set of nonlinear equations,
namely,
¢~.t =f({nr, t: r ~ 1, 2 . . . . . R, ! ffi l, 2 . . . . . L}),
(4.2)
w~zffipr, t ( l + n z - ~ E ) ~ t ) , : . ,
(4.3)
K~
A~ =
(4.4)
Xl~S(r) W r,
* l,
n: -~ ~
Arw~t,
(4.5)
r~R(l)
where for conciseness we have omitted the argument K. We may solve this set of equations
by cyclic iteration through (4.2) to (4.5).
In order to estimate the correction terms ¢¢,t = m.t(K) - nr.z(K - e:), we make two
assumptions:
(1) Only the chain with one customer removed is affected; i.e.,
¢¢,t == 0
for j # r.
(4.6)
(2) For the affected chain ( j ffi r) the corrections can be estimated by means of a single
coalesced closed chain problem. We write
~.t ffi n,(K,) - Jt(Kr - 1),
(4.7)
Mean-Value Analysis o f Closed Multichain Queuing Networks
321
where ffj is the mean queue size of service center l in a single chain queuing network with
Kr chain r customers and with modified parameters
f pr,tnt
if l ~ S(r),
pl=l~r't
(4.8)
otherwise.
Note that nt and n,,t = )~,w~z are those quantities in effect at a given iteration cycle of
Algorithm 3. The operations count per iteration cycle is of the order L K where K = K1 +
K2 + . . . + KR, dearly an affordable effort even fordarge problems. We shall show that
this heuristic algorithm is asymptotically valid as the population becomes large.
We consider the limit a ~ oo such that Kr = K,a and Kr is fixed, r = 1. . . . . R. The
quantity ¢,',t, given by (4.7) and (4.8), is the increment in mean queue size for a single chain
network when the populaUon size is increased by 1. It can be shown (e.g., [8]) that 0 _<
Err,I ~ 1. Let nt = lim~_~nt(K)/a, ff~l = lim~_,~w~t(K)/a, ~, = lim~_.~h,(K).
We assume that these limits exist for all I. It then follows from (4.3) to (4.6) that
I~r*,l "~ pr, ll~l,
(4.9)
. *
~IES(r) W r,l
nt =
,
~ ~,Wr*,t.
rER(1)
(4.10)
(4.11)
Pittel [10] proved rigorously that (4.9) to (4.11) describe the limit a ~ oo of queuing
networks in the class Q(K). Furthermore, he proved convergence of cyclic iteration through
(4.9) to (4. l 1). The examples given in [12] show that the heuristic is within a few percent
of the true value for rather small population sizes. Further validation work is in progress.
In summary, we showed that a few very simple and intuitively appealing principles
completely govern the mean-value behawor of product-form queuing networks with
multiple closed chains. These principles are
(i) an arriving customer "sees" the system with himself removed in equilibrium;
(ii) Little's equation applied to chains;
(iii) Little's equation applied to service centers.
They allow a very simple and stable calculation of most quantities of interest to modelers.
The three principles yield considerable insight into the analysis of closed multichain
networks. This has led to a heuristic method which allows for the approximate solutton of
problems with very many closed chains. The heurisUc was shown to be asymptotically
valid, and hence it should work better the larger the problem. Large computer communication networks with end-to-end flow control windows are now amenable to analytic
treatment. Another heuristic generalization which covers FCFS service centers with general
chain dependent service demand distributions is currently being investigated.
ACKNOWLEDGMENTS. The authors are grateful for the helpful discussions they had with
R. Wolff, P. J. Schweitzer, and H. Kobayashi.
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RECEIVED JUNE 1978; REVlSED MAY 1979; ACCEPTEDMAY 1979
JournaloftheAssoclaUonforComputmgMachmeTy.Vol 27.No 2, April1980