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Some equivalencies between closed queueing networks with blocking
111
Some Equivalencies between Closed Queueing Networks with Blocking * Raif O. Onvural and H.G. Perros
1. Introduction
Department of Computer Science, Center for Communications and Signal Processing, North Carolina State University, Raleigh, NC 27695-8206, U.S.A.
u s e f u l in m o d e l i n g c o m p u t e r s y s t e m s , d i s t r i b u t e d
Closed queueing networks have proved to be systems, production systems and flexible manufac-
Received October 1987 Revised April 1988
turing systems. In recent years, there has been a g r o w i n g i n t e r e s t in the d e v e l o p m e n t of c o m p u t a tional m e t h o d s for the analysis of q u e u e i n g net-
We obtain equivalencies between closed queueing networks with blocking with respect to buffer capacities and the number of customers in the network. These results can be used in approximations, in the buffer allocation problem as well as to explain the behavior of closed queueing networks with finite buffers.
Keywords: Blocking, Closed Queueing Networks.
w o r k s w i t h f i n i t e b u f f e r s . T h i s is p r i m a r i l y d u e to a g r o w i n g n e e d to m o d e l actual s y s t e m s w h i c h have finite capacity resources. An important
fea-
t u r e o f s y s t e m s w i t h f i n i t e b u f f e r s is t h a t a s e r v e r may become blocked when the capacity limitation o f t h e d e s t i n a t i o n q u e u e is r e a c h e d . V a r i o u s b l o c k ing m e c h a n i s m s have b e e n c o n s i d e r e d in the litera t u r e so far. T h e s e b l o c k i n g m e c h a n i s m s a r o s e o u t o f v a r i o u s s t u d i e s o f r e a l life s y s t e m s . A d i s c u s s i o n on these different blocking mechanisms f o u n d i n [15].
can be
Closed queueing networks with infinite buffer capacities, under
certain restrictions, have been
s h o w n to have p r o d u c t - f o r m
steady state queue
l e n g t h d i s t r i b u t i o n s (cf. [6,8]). I n g e n e r a l , c l o s e d queueing networks with finite buffers (hereafter
Rail. O. Onvural received his B.S. degree in Industrial Engineering from Middle East Technical University, Turkey, in 1981, M.S. in Industrial Engineering and Operations Research from Syracuse University in 1983, M.S. in Computer Studies and Ph.D. in joint Computer Studies/Operations Research from North Carolina State University at Raleigh in 1985 and 1987 respectively. From 1987 to 1988, he was a research associate at the Center for Communications and Signal Processing at NCSU. Currently, he is a member of scientific staff at Bell Northern Research, Research Triangle Park, NC. His current research interests include queueing networks with blocking and performance evaluation of communication networks, distributed systems and ISDN systems. * Supported in part by the National Science Foundation under grant DCR-85-02540. North- Holland Performance Evaluation 9 (1988/89) 111-118
H.G. Perros received the B.Sc. degree in Mathematics in 1970 from Athens University, Greece, the M.Sc. degree in Operational Research with Computing from Leeds University, England, in 1971 and the Ph.D. degree in Operations Research from Trinity College Dublin, Ireland, in 1975. From 1976 to 1982 he was an Assistant Professor in the Department of Quantitative Methods, University of Illinois at Chicago. In 1979 he spent a sabbatical term at INRIA, Rocquencourt, France. In 1982 he joined the Department of Computer Science, North Carolina State University, as an Associate Professor, and since 1988 he is a Professor. He was the conference chairman of PERFORMANCE '86 and ACM SIGMETRICS 1986 conference and he was also the conference co-chairman of the First International Workshop on Queueing Networks with Blocking. His research interests are in the area of computer and communication performance modelling. He is a member of ACM, SIGMETRICS, ORSA, ORSA/TIMS Applied Probability Group, IEEE Computer and Communication Societies and HELORS. Currently, he is on a sabbatical leave of absence first at BNR, Research Triangle Park, North Carolina, and subsequently at the University of Paris 6, France.
0166-5316/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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R.O. Onoural, H.G. Perros / Equivalencies between CQN-B
referred to as CQN-B) could not be shown to have product form solutions, excluding the following special cases: (a) the routing matrix is reversible; (b) the branching probability depends on the state of the originating node and the state of the destination node; (c) the probability of blocking does not depend on the number of customers in the destination node but simply is constant; (d) the service rate at each node is constant and at most one node is allowed to be empty; (e) reversible networks with state dependent transition probabilities (cf. [4,10-12]). Also, a CQN-B has always a product form solution if it consists of two nodes (cf. [3.9]). A survey of closed queueing networks with blocking can be found in [13]. In this paper, we obtain equivalencies between C Q N - B with respect to buffer capacities and the number of customers in the network. Two CQN-B are equivalent if they have the same rate matrix. The equivalencies presented here do not constitute a solution technique for CQN-B. Rather, they provide an insight into the behavior of their performance. We will consider closed queueing networks consisting of N nodes and K customers. Each node consists of a single queue served by a server with an exponentially distributed service time with rate #i. Let B, denote the capacity of node i including the server. A customer upon completion of its service at node i attempts to enter destination node j with probability p~j, i, j = 1 . . . . . N. If, at that moment, node j is full, the customer will be forced to wait in front of server i until a space becomes available at node j. Server i remains blocked for this period of time, and it can not serve any other customer waiting in its queue. If more than one server is blocked by the same node, then these servers will get unblocked in a firstblocked-first-unblocked fashion. Following the classification given in [15] this blocking mechanism will be referred to as a type 1 blocking mechanism. Due to the blocking mechanism described above, and due to the fact that these N nodes are arbitrarily interconnected, it is possible that deadlocks m a y occur. In this paper, it is assumed that deadlocks are detected immediately and resolved by instantaneously exchanging blocking units. The interested reader may refer to [16] for details. Equivalencies between CQN-B with respect to
buffer capacities and the number of customers in the network are presented in Section 2 and conclusions are given in Section 3.
2. CQN-B under type 1 blocking mechanism Let n = n l i n i = 1 N ( B i } . Clearly the number of customers in the network, K, is such that 1 ~< K ~< Eu= 1Bi. For 1 ~< K ~< n, blocking does not occur and hence the network has a product form solution. This product form solution can be obtained by treating the queueing network as if the queue at each node has an infinite capacity (cf. [8]). When the number of customers is greater than the minimum buffer capacity, then blocking will occur. In this case, product f o r m solutions are, in general, not available. Two networks are defined to be equivalent if they have the same rate matrix. In most cases, we will establish these equivalencies by first showing that both networks have the same number of states. Then, we define mappings between the states of two networks such that the transition rates into and out of corresponding states are the same, thus showing that they have the same rate matrix. . . . . .
2.1. A g g r e g a t i o n p r i n c i p l e
In this section, we will introduce a concept called 'aggregation principle', which is used in some of the results presented below. Consider a node j of a C Q N - B under type 1 blocking with buffer capacity Bj. If Bj = K - 1 , where K is the number of customers in the network, then there can be at most one node blocked by node j at any time. In this case, the blocked node has exactly one customer (which has completed its service), node j is full and all other nodes are empty. For states in which node l is blocked by node j, we will use the superscript*, i.e., i~*, to denote that node l is blocked while superscript l, i.e. ij = B}, denotes that node j is full and it is blocking node l. Hence, the state (0 . . . . . it* = 1 . . . . . ij = ( K - 1) l. . . . . 0) denotes that node l is blocked by node j. The global balance equation of this state is given as follows: po/~le(0 . . . . . i , = 1 . . . . . i j : ( K = I~jP(O . . . . , i[ ~ = 1 . . . . . ij = ( K -
1) . . . . . 0) 1) t . . . . . 0)
(1)
R.O. Onoural, H. G. Perros / Equioalencies between CQN-B
where, P ( . ) is the steady state queue length distribution of the C Q N - B under consideration. W e note that the right-hand side of (1) is the rate out of the state in which node j is blocking node l and the left-hand side is the rate into such state. Furthermore, define a state (0 . . . . . ij = Bj* + 1 . . . . . 0) such that P(0 ..... i,=~*
+ 1 . . . . . 0)
N
E
P(0- . . . . . i l * = 1 . . . . . i / ( K - 1 )
t . . . . . 0).
I~ 1 , l ~ j
(2) That is, P ( 0 . . . . . ij = Bj* + 1 . . . . ,0) is the probability that node j is blocking a node. S u m m i n g b o t h sides of (1) over l and using equation (2) we have: N
Y~
ptfl~zP(O . . . . . i / = 1 . . . . . 6 = ( K -
1) . . . . . O)
i= 1, l ~ j
=
. . . . . /, =
+ 1 ..... 0).
(3)
Furthermore, departure from a state where node j is blocking n o d e l is independent of the blocked n o d e l and it can not be blocked. This is because all nodes other than n o d e l are e m p t y and (i) if B~> 1, then there is a space at node l or (ii) if B t = 1, then a deadlock will occur. In this case, we are assuming that this deadlock is detected immediately and resolved b y exchanging the blocked customers. Therefore, a customer at n o d e j joins any node k with rate pjk/~j. Now, let us increase the buffer capacity of n o d e j by one (i.e. let Bj = K ) while keeping all the other parameters the same. Then, there c a n n o t be any n o d e blocked by n o d e j. Writing d o w n the equilibrium equation for the state (0 . . . . . ij = K . . . . . 0) we have N
Y'~ p o g , P ( O . . . . . i, = 1 . . . . . ij = ( K -
1) . . . . . O)
i=1
=
j,(o .....
i,
. . . . . o).
(4)
We note that a customer at node j joins node k with rate Pjkgj, k = 1 , . . . , N; k 4=j. Hence, the aggregated state in the C Q N - B with Bj = K - 1, (i.e. (0 . . . . . ij = Bj* + 1 . . . . . 0), defined by (2)) has the same behavior as the state (0 . . . . . ij = K . . . . . 0) in the C Q N with Bj = K, i.e., equation (4), while keeping all the other parameters the same. This p r o p e r t y will be called the aggregation principle.
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2.2. Some properties of C Q N - B under type 1 blocking We n o w proceed to investigate some of the properties of C Q N - B u n d e r type 1 blocking. This is done mainly through a n u m b e r of theorems presented in this section. In the following theorem we show that when a node becomes blocked, the service space in front of the blocked server behaves like an additional buffer space for the blocking node. This is shown only for the case when there are no customers in the blocked node waiting for service during the blocking period.
2.1. Theorem. Consider a C Q N - B under type 1 blocking as described in Section 1 with buffer capacities B i. Let S = { i: B i = K } be the set of nodes with buffer capacities equal to the number of customers in the network. We shall refer to this network as CQN-B-1. Now, consider another CQNB identical to CQN-B-1 except that the buffer capacity of node j, j ~ S, is reduced by one. We shall refer to this network as CQN-B-2. Then the two networks have the same rate matrix if all the states in which node j is blocking another node are aggregated into one state (i.e. applying the aggregation principle) in CQN-B-2.
Proof. Without loss of generality, let n o d e j in CQN-B-1 be such that Bj = K. Let C Q N - B - 2 be identical to CQN-B-1, except that Bj = K - 1. The only difference between these two networks is that in CQN-B-2, node j m a y cause blocking of a n o d e while node j in C Q N - B - 1 will not cause any blocking. We note that there can be at m o s t one node blocked by node j at any time in CQN-B-2. After the aggregation principle is applied to C Q N B-2, i.e. (2), the two networks have the same rate matrix. The equivalency of the steady state queue length distributions can be summarized as follows: Let pB,=K(.), p B , = K - I ( . ) be the steady state probability distributions of C Q N - B - 1 and C Q N - B - 2 respectively. Then for the states in which n o d e j is not blocking a node, we have = p,,,=K-,(.).
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R.O. Onvural, H.G. Perros / Equivalencies between CQN-B
F o r the states in which n o d e j is blocking node l in CQN-B-2, we have
N P(il,-'-,
N
N
E P + : K - I ( 0 . . . . . i+*=1 . . . . . BJ,0 . . . . . 0)
= £
I=1
=:,="(o
j=l
iN) £
N
•
j=l k=l
PjkSjl'tj
N
£
p/kSkg/P(il .....
. . . . . , , = , , ..... 0).
We note that, in the above proof, it was assumed that there is only one n o d e with Bj = K. If more than one node has the buffer capacity K in C Q N B-l, then the above discussion should be repeated for each one of these nodes. []
--1,...,
2.2. Theorem. Consider a C Q N - B under type 1 blocking as described in Section 1 with buffer capacities B i. I f the number of customers in the network, K, is equal to n + 1, where n = mini= x..... N ( B i } , then the network has a product form solution.
Proof. Let (il, i z . . . . . i n ) be the state of a C Q N - B where ({V is the n u m b e r of customers at node j with E j = a i j = n + l , and ij~
iN) ,
(il . . . . . iN) ~ Z 1, e(o ..... i,=1 ..... ij=Bj ..... 0)pog , = P ( 0 . . . . . i F = 1 . . . . . i2= B/ . . . . . 0)/~,, (0 . . . . . i+*=1 . . . . . 6 = B / . . . . . 0 ) ~ Z ~_,
Let us consider a C Q N - B under type 1 blocking with K = min,=~ ..... N (Bi) + 1 customers in it. In such a network blocking m a y arise when the n o d e with the smallest buffer becomes full. The blocked n o d e will contain exactly one customer which has completed its service and has attempted to enter the full node. Furthermore, all nodes other than the blocked n o d e and the blocking n o d e are empty. I n the following theorem, we prove that a C Q N - B with K = mini=~ ..... N {B,} + 1 customers has a p r o d u c t form solution. We note that this result is immediate from T h e o r e m 2.1. However, in this special case we are able to present the form of p r o d u c t f o r m solution of the blocking network without applying the aggregation principle.
ij + 1 . . . . . i k
k=l
2,
(5)
P(i, ..... iN)=I
(it ..... iN)~Z
where Z 1 and Z 2 denote the set of all feasible states in which no n o d e is blocked and in which one node is blocked respectively. Z = Z 1 tJ Z 2 is the set of all feasible states. Let iN) if ij ~< Bj, j = l
'n'(i I .....
P ( i , . . . . . iN) =
. . . . . N,
PO et :,~ ---~-¢r[v . . . . . ij = K . . . . . O)
(6)
if it* = 1 and ij = BJ where ej is the m e a n n u m b e r of visits a customer makes to the ith n o d e per unit time and ~r(-) is the solution obtained by assuming that C Q N - B has infinite buffers and K = n + 1. Hence, It(.) has a p r o d u c t form solution, i.e. (cf. [8]), N
rr(i, . . . . . iN) = G ( n + 1) 1-I ( e J g j ) i'j=l
Clearly, E(i ...... J u ) e z P ( i l . . . . . iN) = 1. By substituting these expressions for P ( . ) into (5), it can be easily verified that the balance equations are satisfied. In particular, for states where a n o d e is blocked, we have e(o .....
i, = 1 . . . . .
6 = g ..... O)p,:,
= P ( 0 . . . . . i+*=1 . . . . . i , = B J . . . . . 0 ) # / , or
..... =
i,= 1,..., .....
i + = B+ . . . . . O ) p , : ,
i+ = K . . . . .
R.O. Onvural, H.G. Perros /Equivalencies between CQN-B
which can b e verified b y s u b s t i t u t i n g the a b o v e p r o d u c t f o r m s o l u t i o n of ~r(.). [] U n d e r certain restrictions, the rate m a t r i x of a C Q N - B u n d e r t y p e 1 b l o c k i n g is i d e n t i c a l for a r a n g e of values of K. In p a r t i c u l a r , let there be a n o d e m in a C Q N - B with B,, > ( E N = I B i ) - Bm, i.e., the b u f f e r c a p a c i t y of n o d e m is greater t h a n the r e m a i n i n g total c a p a c i t y of the network. T h e n the following t h e o r e m proves t h a t this is a sufficient c o n d i t i o n for the n e t w o r k to have the s a m e rate m a t r i x for a r a n g e of values of K. W e n o t e that C Q N - B in which exactly one n o d e has an infinite c a p a c i t y is a special case which satisfies this c o n d i t i o n . 2.3. T h e o r e m . Consider a C Q N - B under type 1 blocking as described in Section 1 with buffer capacities B i. Let B,, = m a x / = 1 . . . . . N (Bi)" I f Om > (ZN=1Bi) -- Bin, then the network has the same rate matrix for all K ~ S, where S = { L: ( ~ = 1Bi) - B m + 1 <~L <~B m }. Furthermore, the network with K = B m + 1 customers in it has the same rate matrix for all K ~ S if all the states in which node m is blocking another node are aggregated into one state (i.e. if the aggregation principle is applied). Proof. C o n s i d e r all n o d e s o t h e r t h a n n o d e m. W e first n o t e that the d i s t r i b u t i o n o f K c u s t o m e r s over the r e m a i n i n g N - 1 n o d e s is i n d e p e n d e n t of K. This is b e c a u s e K takes values greater than (~N=1B ~) -- Bm, the total c a p a c i t y of the r e m a i n i n g nodes. (Obviously, this will n o t be the case if K < (EN=IBi) -- Bin). In o r d e r to clarify this point, let us c o n s i d e r the states of the system for K > (YU= 1B , ) - B m. T h e r e is a state in which all N - 1 n o d e s are full a n d a n o t h e r state in which all N - 1 n o d e s are empty. T h e r e m a i n i n g states reflect all the p o s s i b l e c o m b i n a t i o n s in b e t w e e n with im = K N - Y~j=1j~,~ij. N o w , let K * = K + k, where K * B,,. T h e n we can easily verify that for K * we still o b t a i n the s a m e states, o n l y i m will c o n t a i n k m o r e customers. Similarly, it can b e easily verified that the n u m b e r of states where s o m e node(s) is b l o c k e d is i n d e p e n d e n t of K for K>~ ( E N = I B i ) B m. Hence, the state space of the N - 1 n o d e s a n d the t r a n s i t i o n s b e t w e e n t h e m are i n d e p e n d e n t of K. F u r t h e r m o r e , for (y.U=aBi) - B,, + 1 ~< K ~< B m n o d e m c a n n o t b l o c k a n y n o d e a n d c a n n o t be e m p t y . Hence, t r a n s i t i o n s into a n d out of n o d e m are i n d e p e n d e n t of K, K ~ S. Therefore, for all
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K ~ S, we have the s a m e rate matrix. T h a t is, for K a n d K * such t h a t ( E N _ 1 B s ) - B m+I~~ ( ~ N = I B i ) - - B m + 1. Then CQN-B-1 with K customers has the same rate matrix as C Q N - B - 2 with K * = K + B * - B m customers. Proof. Let i m a n d z*m be the n u m b e r of c u s t o m e r s at n o d e m in C Q N - B - 1 a n d C Q N - B - 2 respectively. W e first n o t e that, K - ~ N = ~ B , ~ < i m~< min(Bm, K ) in C Q N - B - 1 , a n d K * ENi=IBi ~< z,,* ~< m i n ( B * , K * ) i n C Q N - B - 2 . Hence, im a n d tm'* c a n take the s a m e n u m b e r of (different) values in b o t h networks. F u r t h e r m o r e , given the n u m b e r of customers, ira, in C Q N - B - 1 a n d i* = i m + K * - K in C Q N - B - 2 , n o d e m in C Q N - B - 1 a n d C Q N - B - 2 c a n n o t be e m p t y a n d the n u m b e r of c u s t o m e r s left to be d i s t r i b u t e d over the r e m a i n i n g N - 1 n o d e s are the s a m e in b o t h networks. Similarly, it can be a r g u e d that the n u m b e r of states in which one or m o r e n o d e s are b l o c k e d is the s a m e in b o t h networks. ( W e n o t e that this is n o t the case if n o d e m is allowed to be e m p t y . ) Hence, b o t h n e t w o r k s have the s a m e n u m b e r of states. F u r t h e r m o r e . a -
R.O. Onoural, H.G. Perros /Equiualencies between CQN-B
116
( i I . . . . . i ...... i N ) of C Q N - B - 1 is equivalent to the state (i 1. . . . . i,, + K * - K . . . . . iN) of C Q N B-2 in the sense that b o t h states have the s a m e t r a n s i t i o n rates into a n d out o f c o r r e s p o n d i n g equivalent states. Hence, state
p l ( i 1. . . . . i . . . . . .
iN)
= P Z ( i I . . . . . i,,, + K * - K
.....
iN)
w h e r e p l ( . ) a n d p 2 ( . ) are the s t e a d y state q u e u e length d i s t r i b u t i o n s of C Q N - B - 1 a n d C Q N - B - 2 respectively. [] 2.5. Corollary. Consider a C Q N - B - 1 under type 1 blocking as described in Section 1 with buffer capacities B~. Also, consider a C Q N - B - 2 identical to C Q N - B - 1 but with buffer capacities C,, i = 1 . . . . . N. L e t K 1 and K 2 be the n u m b e r o f customers in C Q N - B - 1 a n d C Q n - B - 2 respectively such that no node can be empty. Then the two networks have the s a m e rate m a t r i x i f they have the s a m e n u m b e r o f free spaces, i.e. (EN=IBi) -- K x = ( E N = I C i ) - - K 2. W e first n o t e that if no n o d e is allowed to be e m p t y , then the n o d e with the m a x i m u m b u f f e r c a p a c i t y c a n n o t b e empty. N o w , let d i = Bi - Cj, i = 1 . . . . . N. T h e n a state (ia . . . . . iN) of C Q N - B - 1 is equivalent to a state ( i ~ - d 1. . . . . i N - dN) of C Q N - B - 2 in the sense t h a t b o t h states have the s a m e t r a n s i t i o n rates into a n d out of corres p o n d i n g equivalent states. Hence, Proof.
Pa( i a . . . . . i N ) = p 2 ( i I -- d x . . . . .
r u p t e d a n d the n o d e is blocked. Service is r e s u m e d f r o m the i n t e r r u p t i o n p o i n t as soon as a d e p a r t u r e occurs f r o m the d e s t i n a t i o n node. Blocking time is ignored. W h i l e the server is b l o c k e d , the p o s i t i o n in front o f the server is o c c u p i e d b y the customer. T h e following t h e o r e m is a n e x t e n s i o n of T h e o r e m 2.3 to t y p e 2.2 blocking.
i N --
dN )
where p l ( . ) a n d p 2 ( . ) are the s t e a d y state q u e u e length d i s t r i b u t i o n s of C Q N - B - 1 a n d C Q N - B - 2 respectively. [] N o w , we will discuss the a p p l i c a b i l i t y o f the results p r e s e n t e d a b o v e for t y p e 1 b l o c k i n g to C Q N - B u n d e r a n o t h e r b l o c k i n g m e c h a n i s m first i n t r o d u c e d b y G o r d o n a n d Newell [7]. A c c o r d i n g to the classification given in [15], this b l o c k i n g m e c h a n i s m will b e referred to as type 2.2 b l o c k ing. I n a t y p e 2.2 b l o c k i n g m e c h a n i s m , a c u s t o m e r in q u e u e i declares its d e s t i n a t i o n queue j j u s t b e f o r e its starts its service. If queue j is full, then the i t h server b e c o m e s blocked. W h e n a d e p a r t u r e occurs f r o m d e s t i n a t i o n queue j , the i th server b e c o m e s u n b l o c k e d a n d the c u s t o m e r begins receiving service. If the d e s t i n a t i o n n o d e b e c o m e s full d u r i n g the service, then the service is inter-
2.6. Theorem. Consider a deadlock-free C Q N - B under type 2.2 blocking with buffer capacities B i a n d let M = EN=1Bi be the total capacity o f the network. N o w , /et B m = m a x / = 1..... N ( B i ) . Then, i f B m > M - B,,, then the n e t w o r k has the s a m e rate m a t r i x f o r all K ~ S, where S = { L: M - B m <~L
<.Bin). W e n o t e that the set S, given in T h e o r e m 2.6 is different f r o m the set given in T h e o r e m 2.3. In b o t h theorems, the n u m b e r of c u s t o m e r s is such that the t r a n s i t i o n s b e t w e e n n o d e s (excluding n o d e m ) are i n d e p e n d e n t of K. Hence, the limits are d e t e r m i n e d b y the t r a n s i t i o n s into a n d o u t of n o d e m. I n p a r t i c u l a r , in T h e o r e m 2.3, the e q u i v a l e n c y of the n e t w o r k with K = B , , and K=B m+l c u s t o m e r s is o b t a i n e d using T h e o r e m 2.1. This t h e o r e m is n o t a p p l i c a b l e to n e t w o r k s u n d e r t y p e 2.2 blocking. F u r t h e r m o r e , it can b e easily shown t h a t a n e t w o r k u n d e r t y p e 2.2 b l o c k i n g d o e s n o t have the s a m e n u m b e r of states with K = B m a n d K = B m + 1 customers. Also, for n e t w o r k s u n d e r t y p e 2.2 blocking, it is p o s s i b l e that n o d e m is e m p t y for the values in the set S. This is n o t the case for n e t w o r k s u n d e r t y p e 1 blocking. This is d u e to the difference in the d e f i n i t i o n s of the two b l o c k i n g m e c h a n i s m s . In t y p e 1, b l o c k i n g occurs after service c o m p l e t i o n , while in t y p e 2.2, b l o c k ing occurs b e f o r e service starts. I n p a r t i c u l a r , in t y p e 1 blocking, the state (i I = B 1. . . . . i r a = 0 . . . . . i s = B u ) of the n e t w o r k with K = M - B m c u s t o m e r s is n o t equivalent to the state (i I = B 1 , . . . , i m = 1 . . . . , i s = B N), since the total rates o u t ot these two states are n o t the same. F u r t h e r more, it c a n be easily shown that the total n u m b e r of states in the two n e t w o r k s is n o t the same. In t y p e 2.2 blocking, the total rates o u t of the a b o v e two states are the same, since there c a n n o t be a n y d e p a r t u r e f r o m n o d e m in either case. G i v e n these differences, T h e o r e m 2.6 c a n b e p r o v e d following similar a r g u m e n t s as in T h e o r e m 2.3. Finally, we n o t e that T h e o r e m 2.4 given a b o v e for a t y p e 1 b l o c k i n g m e c h a n i s m is also a p p l i c a b l e
R.O. Onvural, H.G. Perros / Equivalencies between CQN-B
to deadlock-flee CQN-B under a type 2.2 blocking mechanism. 2.3. Some applications of the equivalencies between
CQN-B We now proceed to demonstrate how some of these theorems can be used. Let us first define ~ , ( K ) and L~(K) to be the throughput and the mean queue length of node i respectively with K customers in the network. Furthermore, let X ( K ) denote the throughput of the network. Then, ~.(K)e,-- ~ ( K ) , i = 1 , . . . , N, where e i is therelative number of visits a customer makes to the i th node per unit time. For presentation purpose, let us consider a three node CQN-B under type 1 blocking with buffer capacities 12,5,3. We shall refer to this network as CQN-B-1. From Theorem 2.3, CQN-B-1 has the same rate matrix for K = 9 , . . . , 13. Hence, from the equivalency of the states of CQN-B-1 with K = 10, 11, 12 customers, we have L 1 ( K ) = I + L 1 ( K - 1 ) , while L2(K ) and L 3 ( K ) are the same for K = 9 . . . . . 12. Clearly, the throughput of the network is the same for all K = 9 . . . . . 13. Now, let Xm~ = maxK=l ..... M ~ ( K ) be the maximum throughput of the network w.r.t. K, where M is the total capacity of the network, i.e. M = B 1 + 8. If B 1 >~ 9, then the value of ~kma x is the same for 9 ~< K~< B1 + 1. Now, consider a CQN-B identical to CQN-B1 with buffer capacities 2,2,2. We shall refer to this network as CQN-B-2. F r o m Corollary 2.5, CQN-B-1 with 19 customers has the same rate matrix as CQN-B-2 with 5 customers. Then, Ll~(19) = 10 + L~(5), L~(19) = 3 + L22(5), and L~(19) = 1 +L3~(5), where L~(19) and L2(5) are the mean queue lengths of node i in CQN-B-1 and CQN-B-2 respectively. Now, let us consider two three-node-CQN-B with identical parameters and buffer capacities 4,3,2 with 6 customers and 5,3,2 with 7 customers. Then, from Theorem 2.4 the two networks have the same rate matrix, and, hence, L2(7) = 1 + Ll,(6), L22(7) = L~ (6),
117
studied by Suri and Diehl [19,20], Dallery and Frein [7], and Onvural and Perros [16]. Such networks meet the condition of Theorem 2.3. Theorem 2.3 is also used in [17] to obtain equivalencies between open and closed queueing networks with blocking. Using the monotonicity of the throughput w.r.t, the number of customers in the network (cf. [18]), this theorem gives the number of customers at which the throughput is maximal (w.r.t. K), which is also the throughput of an equivalent OQN-B. An upper bound on the throughput of OQN-B is easily established using Theorem 2.3. Theorem 2.2 presents a product form solution for CQN-B under type 1 blocking with K = min(Bg) + 1. It was also shown that this is the lower bound of the throughput of O Q N - B which is tighter than the bound given with K = min(Bg). The interested reader may refer to [17] for further details. In some of the approximations developed for CQN-B, the analysis is based on a mapping from the states of the network under study to the states of the same network with infinite buffer capacities (cf. [1,2,19,20]). For CQN-B which meets the condition of Theorem 2.1, 2.3, 2.4 or 2.6, one can use these theorems to find an equivalent C Q N - B with smaller buffer capacities a n d / o r fewer customers which may be easier to analyze, or, it may yield better approximations. For presentation purposes, consider a CQN-B with (B 1, B 2, B 3, B 4 ) = (10, 10, 10, 10) and K = 37. To calculate the mean queue lengths, Akyildiz [2] uses a mapping from the states of a closed queueing network with infinite capacities and with K = 37 customers to the states of the CQN-B. We note that this is the only algorithm reported in the literature to calculate the mean queue lengths of CQN-B under type 1 blocking with arbitrary topologies. For this example, the network with infinite buffer capacities has 7770 states. Now, consider the equivalent C Q N - B with (B1, B2, B3, 9 4 ) = (4, 4, 4, 4) and K = 13 which is obtained using Corollary 2.5. In this case, the algorithm required a mapping with K = 13 customers in which the network with infinite buffer capacities has 286 states. Hence, the computational effort is significantly reduced.
L 2 (7) = L~ (6), Xl, (7) = ~] (6)
for i = 1, 2, 3.
CQN-B under type 1 and 2.2 blocking with exactly one node with infinite capacity has been
3. Conclusions
We presented some equivalencies between CQN-B under two types of blocking mechanisms.
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R.O. Onvural, H.G. Perros /Equivalencies between CQN-B
These results are also applicable to CQN-B under some other blocking mechanisms through equivalencies between different blocking mechanisms (cf. [5,14]). Theorem 2.2 presents a product-form solution for networks under type 1 blocking when the number of customers in the network is equal to the minimum buffer capacity plus one. Theorems 2.1, 2.3, 2.4 and 2.6 do not present a solution technique for CQN-B. However, they provide and insight into the behavior of their performance.
[8] [9]
[10]
[11] [12]
References [1] I.F. Akyildiz, On the exact and approximate throughput analysis of closed queueing networks with blocking, IEEE Trans. Softw. Eng. SE-14 (1) (1988) 62-70. [2] I.F. Akyildiz, Product form approximations for queueing networks with multiple servers and blocking, IEEE Trans. Comput., to appear. [3] I.F. Akyildiz, Exact product form solution for queueing networks with blocking, 1EEE Trans. Softw. Eng. SE-14 (4) (1988) 418-429. [4] I.F. Akyildiz and Von Brand, Duality in Open and Closed Markovian Queueing Networks with Rejection Blocking, Tech. Rept. 87-11, Computer Science Department, Louisiana, State University, 1987. [5] S. Balsamo, V. De Nitto Persone and G. Iazeolla, Some Equivalencies of Blocking Mechanisms in Queueuing Networks with Finite Capacity, University Roma-II, Res. Rept. # R-86.02, 1986. [6] F. Baskett, K.M. Chandy, R.R. Muntz and J. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. A C M 22 (2) (1985) 249-260. [7] Y. Dallery and Y. Frein, A decomposition method for the approximate analysis of closed queueing networks with blocking, In: H.G. Perros and T. Altiok, Eds, Pre-Conference Proceedings of the First International Workshop on
[13] [14] [15]
[16]
[17]
[18]
[19]
[20]
Queueing Networks with Blocking, NCSU, Raleigh-NC (1988) 201-233. W.J. Gordon and G.F. Newell, Closed queueing systems with exponential servers, Oper. Res. 15 (1967) 254-265. W.J. Gordon and G.F. Newell, Cyclic queueing systems with restricted length queues, Oper. Res. 15 (1967) 266-278. A. Hordijk and N. Van Dijk, Networks of queues with blocking, In: F.J. Kylstra, ed., Performance "81 (North Holland, Amsterdam, 1981) 51-65. F.P. Kelly, The throughput of a series of buffers, Adv. Appl. Probab. 14 (1982) 633-653. L.M. Le Ny, l~tude analytique de reseaux de files d'attante multiclasses h routage variables, RAIRO 14 (1980) 331-347. R.O. Onvural, A survey of Closed Queueing Networks with Blocking, Tech. Rept. CS-87, NCSU, 1987. R.O. Onvural, Closed Queueing Networks with Finite Buffers, Ph.D. Thesis, CSE/OR, NCSU, 1987. R.O. Onvural and H.G. Perros, On equivalencies of blocking mechanisms in queueing networks with blocking, Oper. Res. Lett 5 (6) (1986) 293-298. R.O. Onvural and H.G. Perros, Throughput Analysis of Closed Queueing Networks with Blocking, Tech. Rept. CS87, NCSU, 1987. R.O. Onvural and H.G. Perros, On the equivalencies of open and closed queueing networks with finite buffers, In: Y. Takahashi, Ed., Proc. Internat. Seminar on Performance of Distributed and Parallel Systems, Kyoto, Japan, (1988) 213-228. J.G. Shanthikumar and D.D. Yao, Monotonicity properties in cyclic queueing networks with finite buffers, In: H.G. Perros and T. Altiok, eds, Pre-Conference Proceedings of the First International Workshop on Queueing Networks with Blocking, NCSU, Raleigh, NC (1988) 393-418. R. Suri and G.W. Diehl, A variable buffer size model and its use in analyzing closed queueing networks with blocking, Manage. Sci. 32 (2) (1986) 206-225. R. Suri and G.W. Diehl, A new building block for performance evaluation of queueing networks with finite buffers, A C M Perf. Eoal. Rev. 12 (3) (1984) 134-142.
Some Equivalencies between Closed Queueing Networks with Blocking * Raif O. Onvural and H.G. Perros
1. Introduction
Department of Computer Science, Center for Communications and Signal Processing, North Carolina State University, Raleigh, NC 27695-8206, U.S.A.
u s e f u l in m o d e l i n g c o m p u t e r s y s t e m s , d i s t r i b u t e d
Closed queueing networks have proved to be systems, production systems and flexible manufac-
Received October 1987 Revised April 1988
turing systems. In recent years, there has been a g r o w i n g i n t e r e s t in the d e v e l o p m e n t of c o m p u t a tional m e t h o d s for the analysis of q u e u e i n g net-
We obtain equivalencies between closed queueing networks with blocking with respect to buffer capacities and the number of customers in the network. These results can be used in approximations, in the buffer allocation problem as well as to explain the behavior of closed queueing networks with finite buffers.
Keywords: Blocking, Closed Queueing Networks.
w o r k s w i t h f i n i t e b u f f e r s . T h i s is p r i m a r i l y d u e to a g r o w i n g n e e d to m o d e l actual s y s t e m s w h i c h have finite capacity resources. An important
fea-
t u r e o f s y s t e m s w i t h f i n i t e b u f f e r s is t h a t a s e r v e r may become blocked when the capacity limitation o f t h e d e s t i n a t i o n q u e u e is r e a c h e d . V a r i o u s b l o c k ing m e c h a n i s m s have b e e n c o n s i d e r e d in the litera t u r e so far. T h e s e b l o c k i n g m e c h a n i s m s a r o s e o u t o f v a r i o u s s t u d i e s o f r e a l life s y s t e m s . A d i s c u s s i o n on these different blocking mechanisms f o u n d i n [15].
can be
Closed queueing networks with infinite buffer capacities, under
certain restrictions, have been
s h o w n to have p r o d u c t - f o r m
steady state queue
l e n g t h d i s t r i b u t i o n s (cf. [6,8]). I n g e n e r a l , c l o s e d queueing networks with finite buffers (hereafter
Rail. O. Onvural received his B.S. degree in Industrial Engineering from Middle East Technical University, Turkey, in 1981, M.S. in Industrial Engineering and Operations Research from Syracuse University in 1983, M.S. in Computer Studies and Ph.D. in joint Computer Studies/Operations Research from North Carolina State University at Raleigh in 1985 and 1987 respectively. From 1987 to 1988, he was a research associate at the Center for Communications and Signal Processing at NCSU. Currently, he is a member of scientific staff at Bell Northern Research, Research Triangle Park, NC. His current research interests include queueing networks with blocking and performance evaluation of communication networks, distributed systems and ISDN systems. * Supported in part by the National Science Foundation under grant DCR-85-02540. North- Holland Performance Evaluation 9 (1988/89) 111-118
H.G. Perros received the B.Sc. degree in Mathematics in 1970 from Athens University, Greece, the M.Sc. degree in Operational Research with Computing from Leeds University, England, in 1971 and the Ph.D. degree in Operations Research from Trinity College Dublin, Ireland, in 1975. From 1976 to 1982 he was an Assistant Professor in the Department of Quantitative Methods, University of Illinois at Chicago. In 1979 he spent a sabbatical term at INRIA, Rocquencourt, France. In 1982 he joined the Department of Computer Science, North Carolina State University, as an Associate Professor, and since 1988 he is a Professor. He was the conference chairman of PERFORMANCE '86 and ACM SIGMETRICS 1986 conference and he was also the conference co-chairman of the First International Workshop on Queueing Networks with Blocking. His research interests are in the area of computer and communication performance modelling. He is a member of ACM, SIGMETRICS, ORSA, ORSA/TIMS Applied Probability Group, IEEE Computer and Communication Societies and HELORS. Currently, he is on a sabbatical leave of absence first at BNR, Research Triangle Park, North Carolina, and subsequently at the University of Paris 6, France.
0166-5316/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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R.O. Onoural, H.G. Perros / Equivalencies between CQN-B
referred to as CQN-B) could not be shown to have product form solutions, excluding the following special cases: (a) the routing matrix is reversible; (b) the branching probability depends on the state of the originating node and the state of the destination node; (c) the probability of blocking does not depend on the number of customers in the destination node but simply is constant; (d) the service rate at each node is constant and at most one node is allowed to be empty; (e) reversible networks with state dependent transition probabilities (cf. [4,10-12]). Also, a CQN-B has always a product form solution if it consists of two nodes (cf. [3.9]). A survey of closed queueing networks with blocking can be found in [13]. In this paper, we obtain equivalencies between C Q N - B with respect to buffer capacities and the number of customers in the network. Two CQN-B are equivalent if they have the same rate matrix. The equivalencies presented here do not constitute a solution technique for CQN-B. Rather, they provide an insight into the behavior of their performance. We will consider closed queueing networks consisting of N nodes and K customers. Each node consists of a single queue served by a server with an exponentially distributed service time with rate #i. Let B, denote the capacity of node i including the server. A customer upon completion of its service at node i attempts to enter destination node j with probability p~j, i, j = 1 . . . . . N. If, at that moment, node j is full, the customer will be forced to wait in front of server i until a space becomes available at node j. Server i remains blocked for this period of time, and it can not serve any other customer waiting in its queue. If more than one server is blocked by the same node, then these servers will get unblocked in a firstblocked-first-unblocked fashion. Following the classification given in [15] this blocking mechanism will be referred to as a type 1 blocking mechanism. Due to the blocking mechanism described above, and due to the fact that these N nodes are arbitrarily interconnected, it is possible that deadlocks m a y occur. In this paper, it is assumed that deadlocks are detected immediately and resolved by instantaneously exchanging blocking units. The interested reader may refer to [16] for details. Equivalencies between CQN-B with respect to
buffer capacities and the number of customers in the network are presented in Section 2 and conclusions are given in Section 3.
2. CQN-B under type 1 blocking mechanism Let n = n l i n i = 1 N ( B i } . Clearly the number of customers in the network, K, is such that 1 ~< K ~< Eu= 1Bi. For 1 ~< K ~< n, blocking does not occur and hence the network has a product form solution. This product form solution can be obtained by treating the queueing network as if the queue at each node has an infinite capacity (cf. [8]). When the number of customers is greater than the minimum buffer capacity, then blocking will occur. In this case, product f o r m solutions are, in general, not available. Two networks are defined to be equivalent if they have the same rate matrix. In most cases, we will establish these equivalencies by first showing that both networks have the same number of states. Then, we define mappings between the states of two networks such that the transition rates into and out of corresponding states are the same, thus showing that they have the same rate matrix. . . . . .
2.1. A g g r e g a t i o n p r i n c i p l e
In this section, we will introduce a concept called 'aggregation principle', which is used in some of the results presented below. Consider a node j of a C Q N - B under type 1 blocking with buffer capacity Bj. If Bj = K - 1 , where K is the number of customers in the network, then there can be at most one node blocked by node j at any time. In this case, the blocked node has exactly one customer (which has completed its service), node j is full and all other nodes are empty. For states in which node l is blocked by node j, we will use the superscript*, i.e., i~*, to denote that node l is blocked while superscript l, i.e. ij = B}, denotes that node j is full and it is blocking node l. Hence, the state (0 . . . . . it* = 1 . . . . . ij = ( K - 1) l. . . . . 0) denotes that node l is blocked by node j. The global balance equation of this state is given as follows: po/~le(0 . . . . . i , = 1 . . . . . i j : ( K = I~jP(O . . . . , i[ ~ = 1 . . . . . ij = ( K -
1) . . . . . 0) 1) t . . . . . 0)
(1)
R.O. Onoural, H. G. Perros / Equioalencies between CQN-B
where, P ( . ) is the steady state queue length distribution of the C Q N - B under consideration. W e note that the right-hand side of (1) is the rate out of the state in which node j is blocking node l and the left-hand side is the rate into such state. Furthermore, define a state (0 . . . . . ij = Bj* + 1 . . . . . 0) such that P(0 ..... i,=~*
+ 1 . . . . . 0)
N
E
P(0- . . . . . i l * = 1 . . . . . i / ( K - 1 )
t . . . . . 0).
I~ 1 , l ~ j
(2) That is, P ( 0 . . . . . ij = Bj* + 1 . . . . ,0) is the probability that node j is blocking a node. S u m m i n g b o t h sides of (1) over l and using equation (2) we have: N
Y~
ptfl~zP(O . . . . . i / = 1 . . . . . 6 = ( K -
1) . . . . . O)
i= 1, l ~ j
=
. . . . . /, =
+ 1 ..... 0).
(3)
Furthermore, departure from a state where node j is blocking n o d e l is independent of the blocked n o d e l and it can not be blocked. This is because all nodes other than n o d e l are e m p t y and (i) if B~> 1, then there is a space at node l or (ii) if B t = 1, then a deadlock will occur. In this case, we are assuming that this deadlock is detected immediately and resolved b y exchanging the blocked customers. Therefore, a customer at n o d e j joins any node k with rate pjk/~j. Now, let us increase the buffer capacity of n o d e j by one (i.e. let Bj = K ) while keeping all the other parameters the same. Then, there c a n n o t be any n o d e blocked by n o d e j. Writing d o w n the equilibrium equation for the state (0 . . . . . ij = K . . . . . 0) we have N
Y'~ p o g , P ( O . . . . . i, = 1 . . . . . ij = ( K -
1) . . . . . O)
i=1
=
j,(o .....
i,
. . . . . o).
(4)
We note that a customer at node j joins node k with rate Pjkgj, k = 1 , . . . , N; k 4=j. Hence, the aggregated state in the C Q N - B with Bj = K - 1, (i.e. (0 . . . . . ij = Bj* + 1 . . . . . 0), defined by (2)) has the same behavior as the state (0 . . . . . ij = K . . . . . 0) in the C Q N with Bj = K, i.e., equation (4), while keeping all the other parameters the same. This p r o p e r t y will be called the aggregation principle.
113
2.2. Some properties of C Q N - B under type 1 blocking We n o w proceed to investigate some of the properties of C Q N - B u n d e r type 1 blocking. This is done mainly through a n u m b e r of theorems presented in this section. In the following theorem we show that when a node becomes blocked, the service space in front of the blocked server behaves like an additional buffer space for the blocking node. This is shown only for the case when there are no customers in the blocked node waiting for service during the blocking period.
2.1. Theorem. Consider a C Q N - B under type 1 blocking as described in Section 1 with buffer capacities B i. Let S = { i: B i = K } be the set of nodes with buffer capacities equal to the number of customers in the network. We shall refer to this network as CQN-B-1. Now, consider another CQNB identical to CQN-B-1 except that the buffer capacity of node j, j ~ S, is reduced by one. We shall refer to this network as CQN-B-2. Then the two networks have the same rate matrix if all the states in which node j is blocking another node are aggregated into one state (i.e. applying the aggregation principle) in CQN-B-2.
Proof. Without loss of generality, let n o d e j in CQN-B-1 be such that Bj = K. Let C Q N - B - 2 be identical to CQN-B-1, except that Bj = K - 1. The only difference between these two networks is that in CQN-B-2, node j m a y cause blocking of a n o d e while node j in C Q N - B - 1 will not cause any blocking. We note that there can be at m o s t one node blocked by node j at any time in CQN-B-2. After the aggregation principle is applied to C Q N B-2, i.e. (2), the two networks have the same rate matrix. The equivalency of the steady state queue length distributions can be summarized as follows: Let pB,=K(.), p B , = K - I ( . ) be the steady state probability distributions of C Q N - B - 1 and C Q N - B - 2 respectively. Then for the states in which n o d e j is not blocking a node, we have = p,,,=K-,(.).
114
R.O. Onvural, H.G. Perros / Equivalencies between CQN-B
F o r the states in which n o d e j is blocking node l in CQN-B-2, we have
N P(il,-'-,
N
N
E P + : K - I ( 0 . . . . . i+*=1 . . . . . BJ,0 . . . . . 0)
= £
I=1
=:,="(o
j=l
iN) £
N
•
j=l k=l
PjkSjl'tj
N
£
p/kSkg/P(il .....
. . . . . , , = , , ..... 0).
We note that, in the above proof, it was assumed that there is only one n o d e with Bj = K. If more than one node has the buffer capacity K in C Q N B-l, then the above discussion should be repeated for each one of these nodes. []
--1,...,
2.2. Theorem. Consider a C Q N - B under type 1 blocking as described in Section 1 with buffer capacities B i. I f the number of customers in the network, K, is equal to n + 1, where n = mini= x..... N ( B i } , then the network has a product form solution.
Proof. Let (il, i z . . . . . i n ) be the state of a C Q N - B where ({V is the n u m b e r of customers at node j with E j = a i j = n + l , and ij~
iN) ,
(il . . . . . iN) ~ Z 1, e(o ..... i,=1 ..... ij=Bj ..... 0)pog , = P ( 0 . . . . . i F = 1 . . . . . i2= B/ . . . . . 0)/~,, (0 . . . . . i+*=1 . . . . . 6 = B / . . . . . 0 ) ~ Z ~_,
Let us consider a C Q N - B under type 1 blocking with K = min,=~ ..... N (Bi) + 1 customers in it. In such a network blocking m a y arise when the n o d e with the smallest buffer becomes full. The blocked n o d e will contain exactly one customer which has completed its service and has attempted to enter the full node. Furthermore, all nodes other than the blocked n o d e and the blocking n o d e are empty. I n the following theorem, we prove that a C Q N - B with K = mini=~ ..... N {B,} + 1 customers has a p r o d u c t form solution. We note that this result is immediate from T h e o r e m 2.1. However, in this special case we are able to present the form of p r o d u c t f o r m solution of the blocking network without applying the aggregation principle.
ij + 1 . . . . . i k
k=l
2,
(5)
P(i, ..... iN)=I
(it ..... iN)~Z
where Z 1 and Z 2 denote the set of all feasible states in which no n o d e is blocked and in which one node is blocked respectively. Z = Z 1 tJ Z 2 is the set of all feasible states. Let iN) if ij ~< Bj, j = l
'n'(i I .....
P ( i , . . . . . iN) =
. . . . . N,
PO et :,~ ---~-¢r[v . . . . . ij = K . . . . . O)
(6)
if it* = 1 and ij = BJ where ej is the m e a n n u m b e r of visits a customer makes to the ith n o d e per unit time and ~r(-) is the solution obtained by assuming that C Q N - B has infinite buffers and K = n + 1. Hence, It(.) has a p r o d u c t form solution, i.e. (cf. [8]), N
rr(i, . . . . . iN) = G ( n + 1) 1-I ( e J g j ) i'j=l
Clearly, E(i ...... J u ) e z P ( i l . . . . . iN) = 1. By substituting these expressions for P ( . ) into (5), it can be easily verified that the balance equations are satisfied. In particular, for states where a n o d e is blocked, we have e(o .....
i, = 1 . . . . .
6 = g ..... O)p,:,
= P ( 0 . . . . . i+*=1 . . . . . i , = B J . . . . . 0 ) # / , or
..... =
i,= 1,..., .....
i + = B+ . . . . . O ) p , : ,
i+ = K . . . . .
R.O. Onvural, H.G. Perros /Equivalencies between CQN-B
which can b e verified b y s u b s t i t u t i n g the a b o v e p r o d u c t f o r m s o l u t i o n of ~r(.). [] U n d e r certain restrictions, the rate m a t r i x of a C Q N - B u n d e r t y p e 1 b l o c k i n g is i d e n t i c a l for a r a n g e of values of K. In p a r t i c u l a r , let there be a n o d e m in a C Q N - B with B,, > ( E N = I B i ) - Bm, i.e., the b u f f e r c a p a c i t y of n o d e m is greater t h a n the r e m a i n i n g total c a p a c i t y of the network. T h e n the following t h e o r e m proves t h a t this is a sufficient c o n d i t i o n for the n e t w o r k to have the s a m e rate m a t r i x for a r a n g e of values of K. W e n o t e that C Q N - B in which exactly one n o d e has an infinite c a p a c i t y is a special case which satisfies this c o n d i t i o n . 2.3. T h e o r e m . Consider a C Q N - B under type 1 blocking as described in Section 1 with buffer capacities B i. Let B,, = m a x / = 1 . . . . . N (Bi)" I f Om > (ZN=1Bi) -- Bin, then the network has the same rate matrix for all K ~ S, where S = { L: ( ~ = 1Bi) - B m + 1 <~L <~B m }. Furthermore, the network with K = B m + 1 customers in it has the same rate matrix for all K ~ S if all the states in which node m is blocking another node are aggregated into one state (i.e. if the aggregation principle is applied). Proof. C o n s i d e r all n o d e s o t h e r t h a n n o d e m. W e first n o t e that the d i s t r i b u t i o n o f K c u s t o m e r s over the r e m a i n i n g N - 1 n o d e s is i n d e p e n d e n t of K. This is b e c a u s e K takes values greater than (~N=1B ~) -- Bm, the total c a p a c i t y of the r e m a i n i n g nodes. (Obviously, this will n o t be the case if K < (EN=IBi) -- Bin). In o r d e r to clarify this point, let us c o n s i d e r the states of the system for K > (YU= 1B , ) - B m. T h e r e is a state in which all N - 1 n o d e s are full a n d a n o t h e r state in which all N - 1 n o d e s are empty. T h e r e m a i n i n g states reflect all the p o s s i b l e c o m b i n a t i o n s in b e t w e e n with im = K N - Y~j=1j~,~ij. N o w , let K * = K + k, where K * B,,. T h e n we can easily verify that for K * we still o b t a i n the s a m e states, o n l y i m will c o n t a i n k m o r e customers. Similarly, it can b e easily verified that the n u m b e r of states where s o m e node(s) is b l o c k e d is i n d e p e n d e n t of K for K>~ ( E N = I B i ) B m. Hence, the state space of the N - 1 n o d e s a n d the t r a n s i t i o n s b e t w e e n t h e m are i n d e p e n d e n t of K. F u r t h e r m o r e , for (y.U=aBi) - B,, + 1 ~< K ~< B m n o d e m c a n n o t b l o c k a n y n o d e a n d c a n n o t be e m p t y . Hence, t r a n s i t i o n s into a n d out of n o d e m are i n d e p e n d e n t of K, K ~ S. Therefore, for all
115
K ~ S, we have the s a m e rate matrix. T h a t is, for K a n d K * such t h a t ( E N _ 1 B s ) - B m+I~
R.O. Onoural, H.G. Perros /Equiualencies between CQN-B
116
( i I . . . . . i ...... i N ) of C Q N - B - 1 is equivalent to the state (i 1. . . . . i,, + K * - K . . . . . iN) of C Q N B-2 in the sense that b o t h states have the s a m e t r a n s i t i o n rates into a n d out o f c o r r e s p o n d i n g equivalent states. Hence, state
p l ( i 1. . . . . i . . . . . .
iN)
= P Z ( i I . . . . . i,,, + K * - K
.....
iN)
w h e r e p l ( . ) a n d p 2 ( . ) are the s t e a d y state q u e u e length d i s t r i b u t i o n s of C Q N - B - 1 a n d C Q N - B - 2 respectively. [] 2.5. Corollary. Consider a C Q N - B - 1 under type 1 blocking as described in Section 1 with buffer capacities B~. Also, consider a C Q N - B - 2 identical to C Q N - B - 1 but with buffer capacities C,, i = 1 . . . . . N. L e t K 1 and K 2 be the n u m b e r o f customers in C Q N - B - 1 a n d C Q n - B - 2 respectively such that no node can be empty. Then the two networks have the s a m e rate m a t r i x i f they have the s a m e n u m b e r o f free spaces, i.e. (EN=IBi) -- K x = ( E N = I C i ) - - K 2. W e first n o t e that if no n o d e is allowed to be e m p t y , then the n o d e with the m a x i m u m b u f f e r c a p a c i t y c a n n o t b e empty. N o w , let d i = Bi - Cj, i = 1 . . . . . N. T h e n a state (ia . . . . . iN) of C Q N - B - 1 is equivalent to a state ( i ~ - d 1. . . . . i N - dN) of C Q N - B - 2 in the sense t h a t b o t h states have the s a m e t r a n s i t i o n rates into a n d out of corres p o n d i n g equivalent states. Hence, Proof.
Pa( i a . . . . . i N ) = p 2 ( i I -- d x . . . . .
r u p t e d a n d the n o d e is blocked. Service is r e s u m e d f r o m the i n t e r r u p t i o n p o i n t as soon as a d e p a r t u r e occurs f r o m the d e s t i n a t i o n node. Blocking time is ignored. W h i l e the server is b l o c k e d , the p o s i t i o n in front o f the server is o c c u p i e d b y the customer. T h e following t h e o r e m is a n e x t e n s i o n of T h e o r e m 2.3 to t y p e 2.2 blocking.
i N --
dN )
where p l ( . ) a n d p 2 ( . ) are the s t e a d y state q u e u e length d i s t r i b u t i o n s of C Q N - B - 1 a n d C Q N - B - 2 respectively. [] N o w , we will discuss the a p p l i c a b i l i t y o f the results p r e s e n t e d a b o v e for t y p e 1 b l o c k i n g to C Q N - B u n d e r a n o t h e r b l o c k i n g m e c h a n i s m first i n t r o d u c e d b y G o r d o n a n d Newell [7]. A c c o r d i n g to the classification given in [15], this b l o c k i n g m e c h a n i s m will b e referred to as type 2.2 b l o c k ing. I n a t y p e 2.2 b l o c k i n g m e c h a n i s m , a c u s t o m e r in q u e u e i declares its d e s t i n a t i o n queue j j u s t b e f o r e its starts its service. If queue j is full, then the i t h server b e c o m e s blocked. W h e n a d e p a r t u r e occurs f r o m d e s t i n a t i o n queue j , the i th server b e c o m e s u n b l o c k e d a n d the c u s t o m e r begins receiving service. If the d e s t i n a t i o n n o d e b e c o m e s full d u r i n g the service, then the service is inter-
2.6. Theorem. Consider a deadlock-free C Q N - B under type 2.2 blocking with buffer capacities B i a n d let M = EN=1Bi be the total capacity o f the network. N o w , /et B m = m a x / = 1..... N ( B i ) . Then, i f B m > M - B,,, then the n e t w o r k has the s a m e rate m a t r i x f o r all K ~ S, where S = { L: M - B m <~L
<.Bin). W e n o t e that the set S, given in T h e o r e m 2.6 is different f r o m the set given in T h e o r e m 2.3. In b o t h theorems, the n u m b e r of c u s t o m e r s is such that the t r a n s i t i o n s b e t w e e n n o d e s (excluding n o d e m ) are i n d e p e n d e n t of K. Hence, the limits are d e t e r m i n e d b y the t r a n s i t i o n s into a n d o u t of n o d e m. I n p a r t i c u l a r , in T h e o r e m 2.3, the e q u i v a l e n c y of the n e t w o r k with K = B , , and K=B m+l c u s t o m e r s is o b t a i n e d using T h e o r e m 2.1. This t h e o r e m is n o t a p p l i c a b l e to n e t w o r k s u n d e r t y p e 2.2 blocking. F u r t h e r m o r e , it can b e easily shown t h a t a n e t w o r k u n d e r t y p e 2.2 b l o c k i n g d o e s n o t have the s a m e n u m b e r of states with K = B m a n d K = B m + 1 customers. Also, for n e t w o r k s u n d e r t y p e 2.2 blocking, it is p o s s i b l e that n o d e m is e m p t y for the values in the set S. This is n o t the case for n e t w o r k s u n d e r t y p e 1 blocking. This is d u e to the difference in the d e f i n i t i o n s of the two b l o c k i n g m e c h a n i s m s . In t y p e 1, b l o c k i n g occurs after service c o m p l e t i o n , while in t y p e 2.2, b l o c k ing occurs b e f o r e service starts. I n p a r t i c u l a r , in t y p e 1 blocking, the state (i I = B 1. . . . . i r a = 0 . . . . . i s = B u ) of the n e t w o r k with K = M - B m c u s t o m e r s is n o t equivalent to the state (i I = B 1 , . . . , i m = 1 . . . . , i s = B N), since the total rates o u t ot these two states are n o t the same. F u r t h e r more, it c a n be easily shown that the total n u m b e r of states in the two n e t w o r k s is n o t the same. In t y p e 2.2 blocking, the total rates o u t of the a b o v e two states are the same, since there c a n n o t be a n y d e p a r t u r e f r o m n o d e m in either case. G i v e n these differences, T h e o r e m 2.6 c a n b e p r o v e d following similar a r g u m e n t s as in T h e o r e m 2.3. Finally, we n o t e that T h e o r e m 2.4 given a b o v e for a t y p e 1 b l o c k i n g m e c h a n i s m is also a p p l i c a b l e
R.O. Onvural, H.G. Perros / Equivalencies between CQN-B
to deadlock-flee CQN-B under a type 2.2 blocking mechanism. 2.3. Some applications of the equivalencies between
CQN-B We now proceed to demonstrate how some of these theorems can be used. Let us first define ~ , ( K ) and L~(K) to be the throughput and the mean queue length of node i respectively with K customers in the network. Furthermore, let X ( K ) denote the throughput of the network. Then, ~.(K)e,-- ~ ( K ) , i = 1 , . . . , N, where e i is therelative number of visits a customer makes to the i th node per unit time. For presentation purpose, let us consider a three node CQN-B under type 1 blocking with buffer capacities 12,5,3. We shall refer to this network as CQN-B-1. From Theorem 2.3, CQN-B-1 has the same rate matrix for K = 9 , . . . , 13. Hence, from the equivalency of the states of CQN-B-1 with K = 10, 11, 12 customers, we have L 1 ( K ) = I + L 1 ( K - 1 ) , while L2(K ) and L 3 ( K ) are the same for K = 9 . . . . . 12. Clearly, the throughput of the network is the same for all K = 9 . . . . . 13. Now, let Xm~ = maxK=l ..... M ~ ( K ) be the maximum throughput of the network w.r.t. K, where M is the total capacity of the network, i.e. M = B 1 + 8. If B 1 >~ 9, then the value of ~kma x is the same for 9 ~< K~< B1 + 1. Now, consider a CQN-B identical to CQN-B1 with buffer capacities 2,2,2. We shall refer to this network as CQN-B-2. F r o m Corollary 2.5, CQN-B-1 with 19 customers has the same rate matrix as CQN-B-2 with 5 customers. Then, Ll~(19) = 10 + L~(5), L~(19) = 3 + L22(5), and L~(19) = 1 +L3~(5), where L~(19) and L2(5) are the mean queue lengths of node i in CQN-B-1 and CQN-B-2 respectively. Now, let us consider two three-node-CQN-B with identical parameters and buffer capacities 4,3,2 with 6 customers and 5,3,2 with 7 customers. Then, from Theorem 2.4 the two networks have the same rate matrix, and, hence, L2(7) = 1 + Ll,(6), L22(7) = L~ (6),
117
studied by Suri and Diehl [19,20], Dallery and Frein [7], and Onvural and Perros [16]. Such networks meet the condition of Theorem 2.3. Theorem 2.3 is also used in [17] to obtain equivalencies between open and closed queueing networks with blocking. Using the monotonicity of the throughput w.r.t, the number of customers in the network (cf. [18]), this theorem gives the number of customers at which the throughput is maximal (w.r.t. K), which is also the throughput of an equivalent OQN-B. An upper bound on the throughput of OQN-B is easily established using Theorem 2.3. Theorem 2.2 presents a product form solution for CQN-B under type 1 blocking with K = min(Bg) + 1. It was also shown that this is the lower bound of the throughput of O Q N - B which is tighter than the bound given with K = min(Bg). The interested reader may refer to [17] for further details. In some of the approximations developed for CQN-B, the analysis is based on a mapping from the states of the network under study to the states of the same network with infinite buffer capacities (cf. [1,2,19,20]). For CQN-B which meets the condition of Theorem 2.1, 2.3, 2.4 or 2.6, one can use these theorems to find an equivalent C Q N - B with smaller buffer capacities a n d / o r fewer customers which may be easier to analyze, or, it may yield better approximations. For presentation purposes, consider a CQN-B with (B 1, B 2, B 3, B 4 ) = (10, 10, 10, 10) and K = 37. To calculate the mean queue lengths, Akyildiz [2] uses a mapping from the states of a closed queueing network with infinite capacities and with K = 37 customers to the states of the CQN-B. We note that this is the only algorithm reported in the literature to calculate the mean queue lengths of CQN-B under type 1 blocking with arbitrary topologies. For this example, the network with infinite buffer capacities has 7770 states. Now, consider the equivalent C Q N - B with (B1, B2, B3, 9 4 ) = (4, 4, 4, 4) and K = 13 which is obtained using Corollary 2.5. In this case, the algorithm required a mapping with K = 13 customers in which the network with infinite buffer capacities has 286 states. Hence, the computational effort is significantly reduced.
L 2 (7) = L~ (6), Xl, (7) = ~] (6)
for i = 1, 2, 3.
CQN-B under type 1 and 2.2 blocking with exactly one node with infinite capacity has been
3. Conclusions
We presented some equivalencies between CQN-B under two types of blocking mechanisms.
118
R.O. Onvural, H.G. Perros /Equivalencies between CQN-B
These results are also applicable to CQN-B under some other blocking mechanisms through equivalencies between different blocking mechanisms (cf. [5,14]). Theorem 2.2 presents a product-form solution for networks under type 1 blocking when the number of customers in the network is equal to the minimum buffer capacity plus one. Theorems 2.1, 2.3, 2.4 and 2.6 do not present a solution technique for CQN-B. However, they provide and insight into the behavior of their performance.
[8] [9]
[10]
[11] [12]
References [1] I.F. Akyildiz, On the exact and approximate throughput analysis of closed queueing networks with blocking, IEEE Trans. Softw. Eng. SE-14 (1) (1988) 62-70. [2] I.F. Akyildiz, Product form approximations for queueing networks with multiple servers and blocking, IEEE Trans. Comput., to appear. [3] I.F. Akyildiz, Exact product form solution for queueing networks with blocking, 1EEE Trans. Softw. Eng. SE-14 (4) (1988) 418-429. [4] I.F. Akyildiz and Von Brand, Duality in Open and Closed Markovian Queueing Networks with Rejection Blocking, Tech. Rept. 87-11, Computer Science Department, Louisiana, State University, 1987. [5] S. Balsamo, V. De Nitto Persone and G. Iazeolla, Some Equivalencies of Blocking Mechanisms in Queueuing Networks with Finite Capacity, University Roma-II, Res. Rept. # R-86.02, 1986. [6] F. Baskett, K.M. Chandy, R.R. Muntz and J. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. A C M 22 (2) (1985) 249-260. [7] Y. Dallery and Y. Frein, A decomposition method for the approximate analysis of closed queueing networks with blocking, In: H.G. Perros and T. Altiok, Eds, Pre-Conference Proceedings of the First International Workshop on
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