2], Boucherie and van Dijk 4], Henderson and Taylor. 13], Serfozo 21]). The obvious advantage of these product form distributions is their simplicity which make ...
On the trac equations for batch routing queueing networks and stochastic Petri nets Richard J. Boucherie1
CWI P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Matteo Sereno2
Universita di Torino Dipartimento di Informatica, Corso Svizzera 185, 10149 Torino, Italy
Abstract The trac equations are a set of linear equations, which are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the trac equations are therefore of great importance. This note provides a necessary and sucient condition on the structure of the network for a solution of the trac equations to exist. The basis of this structural characterisation is the equivalence between batch routing queueing networks and stochastic Petri nets at the level of the underlying stochastic process. Based on new and known results for stochastic Petri nets, this note shows that the new condition stating that each transition is covered by a minimal closed support T -invariant is necessary and sucient for the existence of a solution for the trac equations for batch routing queueing networks and stochastic Petri nets. 1991 AMS Subject Classi cation: Primary: 60K25, Secondary: 90B22 Keywords: trac equations, closed support T -invariant, queueing network, Petri net, recurrence. 1. Introduction
Performance is an important issue in the design and implementation of real life systems such as computer systems, telecommunication networks, and exible manufacturing systems. In many theoretical and practical studies of performance models involving stochastic e ects, the statistical distribution of items over places is of great interest since most of the performance measures such as throughput and utilization can be derived from this distribution. In particular, analytical results related to the structure of the system are of great importance.
During this research the rst author was visiting CWI with the support of the ERCIM Fellowship Programme. 2 Partially supported by the Italian National Research Council \Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo (Grant N. 92.01563.PF69)" and by the ESPRIT{BRA project No.7269 \QMIPS." 1
1. Introduction
2
For queueing networks an important analytical result is the product form equilibrium distribution for the number of customers at the stations. Product form distributions were found by Jackson [14], and are nowadays known for a fairly wide class of queueing models (e.g., Baskett et al. [2], Boucherie and van Dijk [4], Henderson and Taylor [13], Serfozo [21]). The obvious advantage of these product form distributions is their simplicity which make them easy to use for computational issues as well as for theoretical re ections on performance models involving congestion as a consequence of queueing. The starting point for the analysis of product form queueing networks is the assumption that a solution exists for a set of linear equations known as the trac equations. For networks that do not have a product form equilibrium distribution, approximate results for the mean of performance measures can often be obtained via the construction of equations for these mean values. The mean number of customers in the queues and throughputs at the queues might be obtained from the trac equations. Therefore, also for non-product form queueing networks, the existence of a solution for the trac equations is of interest. For standard queueing networks such as Jackson networks [14] or BCMP networks [2], a solution for the trac equations is guaranteed by the Perron-Frobenius theorem (cf. Seneta [20], chapter 1) when the routing process is irreducible. The problem of existence of a solution for the trac equations is completely solved for these standard networks. For batch routing queueing networks, however, the existence of a positive solution for the trac equations cannot be guaranteed in a similar manner. Necessary conditions are available in the literature (cf. Henderson et al. [11], Corollary 1), but a full characterisation of the structure of batch routing queueing networks necessary and sucient for the existence of a positive solution for the trac equations is not available in the literature. This note provides this characterisation. For batch routing queueing networks results related to the structure of the network are not common in the literature. In contrast, for stochastic Petri nets, which are equivalent to batch routing queueing networks at the level of the underlying stochastic process, a lot of results related to the structure of the net are available (see Murata [19] for a recent survey). Therefore, the results of this note are presented in the Petri net formalism. In the Petri net formalism T -invariants play a fundamental role: T invariants characterise both the structure and the evolution in time of the Petri net. This note introduces a new type of T -invariant, that will be called closed support T invariant. The structure of these closed support T -invariants will be analysed in detail. It will be shown that the assumption that all transitions of the Petri net are covered by closed support T -invariants is necessary and sucient for the existence of a positive solution for the trac equations. As a consequence, the existence of a solution for the trac equations is completely characterised on the basis of the structure of the Petri net. This note therefore bridges the gap between the standard starting assumption that a positive solution exists for the trac equations, and the structure of the Petri
3
2. Model
net justifying this assumption. In section 2 we present the basic Petri net notation. In section 3 we characterise the structure of the Petri net that is necessary and sucient for the existence of a positive solution of the trac equations. This characterisation states that all transitions are covered by closed support T -invariants, a new notion for stochastic Petri nets. In addition, we obtain a sucient condition for the existence of an invariant measure (not necessarily of product form) for the Markov chain describing the Petri net. In section 4 we apply these results to batch routing queueing networks and illustrate the structure required to obtain a positive solution of the trac equations. 2. Model
This section presents the basic de nitions of stochastic Petri nets. For additional results and de nitions, see the recent survey of Murata [19]. The speci c assumptions and de nitions characterising the structure of Petri nets that is necessary and sucient for the existence of a positive solution for the trac equations will be given in section 3.
De nition 2.1 (Marked stochastic Petri net) A marked stochastic Petri net is a 6-tuple, SPN = (P; T; I; O; R; m ), where P = fp ; : : : ; pN g is a nite set of places; T = ft ; : : : ; tM g is a nite set of transitions; P \ T = ; and P [ T 6= ;; I; O : P T ! IN are the input and output functions identifying the relation between the places and the transitions; R = (r(t ); : : : ; r(tM )) is a set of ring rates drawn from exponential distributions; and m is the initial marking. 0
1
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A marking m = (m(n); n = 1; : : : ; N ) of a Petri net is a vector in INN , where m(n) represents the number of tokens at place pn, n = 1; : : : ; N . Distributions associated with di erent transitions are independent, and each transition of the Petri net is due to exactly one transition t 2 T that res. The execution policy of the stochastic Petri net is the race model with age memory (cf. Ajmone Marsan et al. [1]). From I (; ) and O(; ) we obtain the vectors I(t) = (I (t); : : : ; IN (t)), and O(t) = (O (t); : : : ; ON (t)), where Ii(t) = I (pi; t), and Oi(t) = O(pi; t). The vectors I(t) and O(t) are called input and output bags of transition t 2 T , representing the number of tokens needed at the places to re transition t, and the number of tokens released to the places after ring of transition t. A necessary and sucient condition for transition t to be enabled in marking m is that m(n) In(t), n = 1; : : : ; N . If transition t res, then the next state of the Petri net is m0 = m ? I(t) + O(t). Symbolically this will be denoted as m[t > m0. A nite sequence of transitions = t1 t2 t is a nite ring sequence of the Petri net if there exists a sequence of markings m1 ; : : : ; m for which m [t > m +1 , i = 1; : : : ; k ? 1. In this case marking m is reachable from marking m1 by ring , 0
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2. Model
denoted as m1 [ > m . The reachability set M(m ) is a subset of INN and gives all possible markings of the Petri net with initial marking m . A transition t 2 T is live if, no matter what marking has been reached from m , it is possible to ultimately re transition t by progressing through some further ring sequence (Murata [19]), that is, if for all m 2 M(m ) there exists an m0 2 M(m) such that t is enabled in m0, where M(m) is the reachability set of the Petri net with initial marking m. A Petri net is live if all its transitions are live. A Petri net is structurally live if there exists an initial marking m for which the Petri net is live. A marking m 2 M(m ) is a home state if m 2 M(m0) for all m0 2 M(m). The incidence matrix is the N M matrix A with entries A(i; t) = Oi(t) ? Ii(t) describing the change in the number of tokens in place pi if transition t res, i = 1; : : : ; N , t 2 T . A vector is the ring count vector of the ring sequence if (t) equals the number of times transition t occurs in the ring sequence . If m [ > m, then m = m + A , an equation referred to as the state equation for the Petri net. A vector x 2 INM is a T -invariant if x 6= 0, and Ax = 0. From the state equation we obtain that a T -invariant corresponds to a ring sequence that brings a marking back to itself (Murata [19]). The support of a T -invariant x is the set of transitions corresponding to non-zero entries of x, and is denoted by kxk, i.e. kxk = ft 2 T jx(t) > 0g. A T -invariant x is a minimal T -invariant if there is no other T -invariant x0 such that x0(m) x(m) for all m. A support is minimal if no proper nonempty subset of the support is also a support of a T -invariant. From Memmi and Roucairol [17] we obtain that there is a unique minimal T -invariant corresponding to a minimal support (minimal support T -invariant), and any T -invariant can be written as a linear combination of minimal support T -invariants. A vector y 2 INN is a P -invariant (sometimes called S -invariant) if y 6= 0, and yA = 0. De nitions of and results for minimal support etc. are analogous to those for T -invariants. P and T -invariants characterise the structural properties of Petri nets. A P -invariant identi es a set of places such that the weighted sum of the number of tokens distributed over these places remains constant for all marking in the reachability set, e.g., the set of places of a Petri net corresponding to a closed Jackson network is a P -invariant. A T -invariant represents a sequence of transitions that brings a marking back to itself. T -invariants are the basis of the characterisation presented in this note. It can be shown that any live and bounded Petri net is covered by P and T -invariants (Murata [19], Silva [22]). The stochastic process describing the evolution of the Petri net is a continuous-time Markov chain, X, with state space isomorphic to the reachability set, that is with state space M(m ) (Molloy [18]). The transition rates of this Markov chain are denoted by Q = (q(m; m0); m; m0 2 M(m )). A collection of positive numbers, m = (m(m); m 2 M(m )), is called an invariant measure if it satis es the global balance equations (Kelly [15], p. 3), 0
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3. The trac equations
X fm(m)q(m; m0) ? m(m0)q(m0; m)g = 0; m02M(m0)
m 2 M(m ):
(2.1)
0
When m is a proper distribution over M(m ) it will be called an equilibrium distribution, and will be denoted by = ((m); m 2 M(m )). As the Markov chain is chosen such that it describes the evolution of the stochastic Petri net under consideration, irreducibility and positive recurrence properties necessary to obtain a unique equilibrium distribution for the Markov chain should be characterised directly from the Petri net structure. (Note that recurrence of the Markov chain is called reversibility in the Petri net literature (Murata [19], p. 548). The notion of reversibility for Petri nets should not be confused with the notion of reversibility for Markov chains (Kelly [15], p. 5).) 0
0
3. The traffic equations
Without loss of generality, we may assume that the ring rate associated with transition t 2 T with input bag I(t) and output bag O(t) can be written as r(t) = (t)p(I(t); O(t)), a form chosen in accordance with the literature on product form results (e.g., Jackson [14], Baskett et al. [2]). Assume that the stochastic Petri net can be represented by a stable and regular, continuous-time Markov chain X = fX (t); t 0g at state space M(m ). Then the transition rates of X are 0
q(I(t); O(t); m ? I(t)) = (t) (m (?mI)(t)) p(I(t); O(t));
(3.1)
for all t 2 T , m 2 M(m ) such that m(n) In(t), n = 1; : : : ; N . We have added the marking dependent function (m ? I(t))= (m) in the transition rates to allow for more generality. For stochastic Petri nets this might seem unnecessary, but for batch routing queueing networks (m ? I(t))= (m) represents the state-dependent service-rate for servicing a group of customers I(t) in state m. The transition rate q(I(t); O(t); m?I(t)) is associated with transition t bringing m to m ? I(t) + O(t). The incidence matrix A of the Petri net is determined by the input and output bags I(t), O(t), t 2 T , of the transitions, and is marking independent. Therefore, a change in (m ? I(t))= (m) 0 does not a ect P A. The total transition rate from m to m = m ? I(t) + O(t) is 0 q(m; m ) = 0 q (I(t); O(t); n). 0
fn; t2T :n+I(t)=m; n+O(t)=m g
Let x ; : : : ; xh denote the minimal support T -invariants found from the incidence matrix. The following de nition and assumption are essential for the analysis presented in this note. Closedness of T -invariants was rst de ned by Donatelli and Sereno [8] as a unifying principle to obtain product form distributions for stochastic Petri nets. A necessary condition for a product form equilibrium distribution similar to closedness is presented in Henderson et al. [11], Corollary 1. 1
6
3. The trac equations
De nition 3.1 (Closed set) For T ST de ne R(T ), the set of input and output bags for the transitions in T , as R(T ) = t2T fI(t) [ O(t)g. T is a closed set if for any g 2 R(T ) there exist t; t0 2 T such that g = I(t), as well as g = O(t0 ), that is if each output bag is also an input bag for a transition in T . The following assumption is the key to the analysis of Petri nets presented in this paper. It guarantees the existence of a positive solution of the trac equations, and implies liveness of the Petri net.
Assumption 3.2 (Minimal closed support T -invariants) Assume that all transitions t 2 T are covered by minimal closed support T -invariants, that is assume that for all t 2 T there exists an i 2 f1; : : : ; hg such that t 2 kxik and kxik is a closed set. Observe that the essential part of the assumption is that all transitions are contained in a closed support. The assumption that all transitions are covered by minimal support T -invariants (closed or not closed) is a natural assumption if we are interested in the equilibrium or stationary distribution of a stochastic Petri net. If this assumption is not satis ed, then there exists a transition, say t , that is enabled in a reachable marking m, and t 62 Shi kxik (if t is never enabled, then we can delete t from T ). Let t re in marking m. Then there exists no ring sequence from m ? I(t ) + O(t ) back to m (otherwise t would be contained in a T -invariant). Thus m is a transient state and does not appear in the equilibrium description of the stochastic Petri net. As a consequence, both m and t can be deleted from the equilibrium description of the Petri net. We now proceed with a characterisation of minimal closed support T -invariants. This shows the relation between minimal closed support T -invariants and `task sequences' (corresponding to a number of tasks that must be executed consecutively) as introduced by Lazar and Robertazzi [16]. 0
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Theorem 3.3 Assume that x is a minimal closed support T -invariant. Then the ring sequence of x is `linear', that is for each t 2 kxk there is a unique t0 2 kxk such that O(t) = I(t0). As a consequence x(i) 1, i = 1; : : : ; M . Conversely, if the ring sequence of a T -invariant x is linear, then x is a closed support T -invariant. Proof Let t 2 kxk. The existence of t0 2 kxk such that O(t) = I(t0) follows from the closedness of kxk. To proof the unicity, let t 2 kxk, and t0 ; t00 2 kxk such that O(t) = I(t0) = I(t00), O(t0) =6 O(t00 ). As a consequence there exists a place p = pi such that maxfOi(t0 ); Oi(t00 )g ? minfOi(t0 ); Oi(t00 )g = 6 0. Without loss of generality, assume that Oi(t0 ) > Oi(t00 ). From the closedness of kxk we obtain that there exist two distinct transitions, say 0t 2 kxk, and t00 2 kxk such that O(t0 ) = I(t0 ), and O(t00 ) = I(t00 ), and we must have one of the three situations depicted in Figure 1: 1
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3. The trac equations t0
t00
t0
p
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t0
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t01
t001
t00
p
t01
t001
t01
t001 p0
p0
Figure 1: (a)
(b)
(c)
(a) O(t0 ) = O(t00). This implies that there exist two ring sequences within kxk that can re independently from I(t) to O(t0 ), in contrast with the assumption that x is a minimal T -invariant. (b) 9 p0 = pj such that maxfOj (t0 ); Oj (t00 )g ? minfOj (t0 ); Oj (t00 )g =6 0, and Oj (t0 ) > 1
1
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Oj (t00 ). This is the situation observed when we considered t0 and t00 and is either followed by situation (a), (b), or (c). (c) as (b), but now Oj (t0 ) > Oj (t00). It is obvious that this is followed by (a), (b), or (c) as well. 1
1
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Finally, since x is a T -invariant, it must be that the ring sequences starting with t0 and t00, say t0 t0 t0 0 and t00t00 t00 00 , are such that O(t0 0 ) = O(t00 00 ) for some 0, 00, that is situation (a) must occur nally, which contradicts the assumption that x is a minimal T -invariant. This establishes unicity. Unicity implies that each transition t 2 x can occur at most once in the ring sequence associated with x, i.e. that x(i) 1, i = 1; : : : ; M . If the ring sequence of a T -invariant x is linear, then for each t 2 kxk there exist s; s0 2 kxk such that O(s) = I(t), O(t) = I(s0) implying that x has closed support. 2 The set of transitions of the Petri net can be decomposed into equivalence classes. This decomposition is based on the set of minimal closed support T -invariants. The equivalence relation is de ned by analogy with a similar equivalance relation presented in Frosch [9], Frosch and Natarajan [10] for cyclic state machines. Assume that the minimal support T -invariants x ; : : : ; xh are numbered such that ClT = fx ; : : : ; xk g is the set of minimal closed support T -invariants (k h). 1
1
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def
1
De nition 3.4 (Common input bag relation) Let x, x0 2 ClT . We say that x, x0 are in common input bag relation (notation: x CI x0 ) if there exist t 2 kxk, t0 2 kx0 k such that I(t) = I(t0 ). The relation CI is the transitive closure of CI . The transitive closure of a relation is de ned as follows: if x, x0 , x00 2 ClT , and x CI x0, x0 CI x00 , then we de ne x CI x0 , x0 CI x00 , x CI x00. This re ects the property and
8
3. The trac equations
that we can go from x to x00 via x0 , and makes the common input bag relation CI an equivalence relation on ClT . Let CI (x) be the equivalence class of x 2 ClT , that is CI (x) = fx0 jx CI x0g. The equivalence classes partition ClT : each x 2 ClT belongs to exactly one equivalence class. As an immediate consequence, under Assumption 3.2, the common input bag relation partitions the transitions t 2 T into equivalence classes too. These equivalence classes correspond to the set of transitions [x0 2CI x kx0 k, as can be seen from the equivalence relation T on the transitions de ned as: tT t0 if there exist x, x0 2 ClT such that t 2 kxk, t0 2 kx0k, and xCI x0. Let T (t) denote the equivalence class of t 2 T . De ne the Markov chain Y = (Y (t); t 0) at nite state space S = fI(t); t 2 T g with transition rates qY (I(t); I(t0)) = (t)p(I(t); I(t0)). This Markov chain Y corresponds to the routing chain as de ned in Henderson et al. [11]. The global balance equations for Y are X fy(I(t))(t)p(I(t); I(t0)) ? y(I(t0))(t0)p(I(t0); I(t))g = 0: (3.2) ( )
t0 2T
It are these global balance equations for Y that correspond to the trac equations for batch routing queueing networks and stochastic Petri nets. De ne S (x) S , the input bags corresponding to CI (x), as
S (x) = fI(t)j9 x0 2 CI (x) such that x0 (t) > 0g: The next theorem shows that the partition of ClT into equivalence classes fCI (x)gx2ClT obtained from the common input bag relation induces a partition fS (x)gx2ClT of S into irreducible sets of the Markov chain Y if and only if Assumption 3.2 is satis ed.
Theorem 3.5 Assumption 3.2 is necessary and sucient for the existence of an invariant measure for Y. Proof Observe that Y has an invariant measure if and only if the state space S
contains irreducible sets only. Therefore it is sucient to prove that Assumption 3.2 is necessary and sucient for the partition of S into irreducible sets fS (x)gx2ClT . Let x; x0 2 ClT . If x0 2 CI (x) then S (x0) = S (x), since CI (x) = CI (x0). If S (x0) \ S (x) 6= ;, then 9 t 2 T such that I(t) 2 S (x0) \ S (x) implying that 9 x00 2 CI (x) for which 9 s 2 T such that x00s > 0 and I(s) = I(t), and 9 x000 2 CI (x0) for which 9 s0 such that x000s0 > 0 and I(s0) = I(t). Thus CI (x00 ) = CI (x000 ) implying CI (x) = CI (x0), in turn implying that S (x0) = S (x). This shows that S (x0) = S (x) if CI (x0) = CI (x), and S (x0) \ S (x) = ; if CI (x0) \ CI (x) = ;. Assumption 3.2 implies that for all t 2 T , 9 x 2 ClT such that t 2 kxk, i.e. 9 S (x) such that I(t) 2 S (x). As a consequence fS (x)gx2ClT forms a partition of S .
9
3. The trac equations
Let I(t); I(t0) 2 S (x). Then 9 x0 ; x00 2 CI (x) for which 9 s; s0 2 T such that x0s > 0 and x00s0 > 0, and I(s) = I(t) and I(s0) = I(t), but also x0CI x00 . Thus 9 , ringsequence, such that I(t)[ > I(t0 ). Let I(t) 2 S (x), I(t0) 2 S (x0), S (x) \ S (x0) = ;. Assume 9 , ring sequence, such that I(t)[ > I(t0) then x0 2 CI (x) implying that S (x) = S (x0). As a consequence fS (x)gx2ClT forms a partition of S into irreducible sets. The Perron-Frobenius theorem (cf. Seneta [20], chapter 1) implies that a positive solution exists to the marking independent trac equations. Conversely, assume that an invariant measure exists to the marking independent trac equations. This immediately implies that for all t 2 T 9 t0 2 T such that O(t) = I(t0). Furthermore, the existence of this invariant measure implies that S is partitioned in irreducible sets. Let Vi, i = 1; : : : ; v, denote the irreducible sets of Y. Let t 2 T and i such that I(t) 2 Vi0 . Since Vi0 is an irreducible set we have that for all v 2 Vi0 9 ; 0 such that I(t)[ > v, and v[0 > I(t). Thus ~ = 0 is a closed support T -invariant. Similarly, from the irreducibility we may conclude that all T -invariants contained in Vi0 have closed support. From Memmi and Roucairol [17] we obtain that each support of an invariant can be decomposed into a union of minimal supports which implies that t is covered by a minimal closed support T -invariant. 2 In the literature, one usually assumes that a solution for the trac equations exists, and necessary conditions are derived from this assumption (e.g., Henderson et al. [11]). Theorem 3.5 provides a necessary and sucient structural condition for the existence of a solution of the trac equations. In example 4.1 we will illustrate the di erence between Assumption 3.2 and the conditions of Henderson et al. [11] that are necessary for the existence of a solution for the trac equations, only. This also shows that Assumption 3.2 is a new condition for the characterisation of product form results. The trac equations for batch routing queueing networks and stochastic Petri nets are closely related to the global balance equations for Y. The trac equations are (recall the form of the transition rates (3.1)) X X y(I(t))(t)p(I(t); O(t0)) = y(I(t0))(t0)p(I(t0); O(t0 )): (3.3) 0
t0 2T
t0 2T :O(t0 )=I(t)
The trac equations balance the average rate at which input bag I(t) is absorbed with the average rate at which input bag I(t) is formed. Observe that this form of balance requires that I(t) is an output bag of a transition t0 if I(t) is an input bag for transition t. The following result states that the trac equations (3.3) are equivalent to the global balance equations (3.2) for Y under Assumption 3.2. Corollary 3.6 (Trac equations) Under Assumption 3.2, y is a positive solution of the trac equations if and only if y is an invariant measure for Y. The following theorem shows that Assumption 3.2 not only guarantees a positive solution for the global balance equations for the routing chain (3.2), but also for the global balance equations (2.1) for the Markov chain X for the stochastic Petri net.
10
3. The trac equations
Theorem 3.7 Under Assumption 3.2, the untimed Petri net (P; T; I; O; m ) underlying the stochastic Petri net SPN has home state m and is structurally live. As a consequence, the stochastic Petri net with ring rates (3.1) is recurrent at M(m ). 0
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Proof Assume that (n) > 0 for all n 2 INN , and (t) > 0 for all t 2 T . This makes 0
the evolution of the stochastic Petri net equivalent to the evolution of the untimed Petri net (as can be seen via the embedded chain at jump moments). Consider the untimed Petri net under Assumption 3.2. 1. Let M (m ) denote the reachability set of the untimed Petri net. Assume that m 2 M (m ) is such that t 2 T is enabled. Then for all t0 2 T (t), the equivalence class of transition t, there exists an m0 2 M(m) such that t0 is enabled in marking m0. The ring sequence from m to m0 can be constructed such that it contains transitions from T only. To see this, observe that for t to be enabled in m it must be that m ? I(t) 2 INN (the enabling condition). Let t 2 kxk, x 2 ClT , and t0 2 T (t). Then there exists an x0 2 ClT such that t0 2 kx0k and xCI x0. As a consequence, there exists a ring sequence from t to t0 , say = t0 t1 t t +1 , t = t0 , t0 = t +1 , such that O(t ) = I(t +1 ), i = 0; : : : ; k. The corresponding sequence of markings is m[t0 > m1 [t1 > m [t > m +1 , where m1 = m ? I(t0 ) + O(t0 ), m2 = m1 ? I(t1 ) + O(t1 ) = m ? I(t0 ) + O(t1 ); : : : ; m = m ? I(t0 ) + O(t ?1 ), m +1 = m ? I(t0 ) + O(t ). Since O(t ) = I(t +1 ) we have that t0 is enabled in m +1 i t is enabled in m. This establishes that if a transition t is enabled then all t0 2 T (t) are live. 2. For all m 2 M (m ) there is a ring sequence such that m [ > m. By induction on the length l of this ring sequence we prove that there is a ring sequence 0 such that m[0 > m . l = 1. Let = t, then m = m ? I(t) + O(t). Assumption 3.2 implies that there exists an x 2 ClT such that t 2 kxk. Let x be the unique linear ring sequence of x (Theorem 3.3), say x = tx 1 tx 2 tx . Without loss of generality, assume that t = tx 1 . Similar to the construction above, if t is enabled then x is enabled, and for 0 = tx 2 tx we have m[0 > m . Assume that for any ring sequence of length k such that m [ > m there is a ring sequence 0 such that m [0 > m . Let l = k + 1 and = t1 t2 t t +1 such that m [ > m. Let = t1 t2 t , m [ > m. It is sucient to prove that there exists a ring sequence such that m[ > m. To this end, observe that there exists an x 2 ClT such that t +1 2 kxk. By the construction used for l = 1 we have that x = t +1 de nes . (Note that this is true because x has closed support.) Now 0 = 0, completing the induction step. As a consequence, m is a home state. 3. Let m be such that at least one transition in each equivalence class T (t) is enabled. Result 2. shows that m is a home state, and result1. implies that all transitions are live. This shows that the untimed Petri net is structurally live. ut
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k
11
3. The trac equations
4. Consider the stochastic Petri net with ring rates (3.1). If (n) > 0 for all n 2 INN then all ring sequences of the untimed Petri net remain possible. We have M(m ) = M (m ) and all m 2 M(m ) are recurrent states (home states). 5. Let m 2 M (m ), and t 2 T such that (m ? I(t)) = 0. From the construction 0
ut
0
0
ut
0
0
of the ring sequence in point 1., we obtain that all ring sequences containing transitions t0 2 T (t) only and passing through m (that is = , and there exists a marking m0 such that m0[ > m[ > m00) are stopped. Conversely, if (m ? I(t)) > 0 then all these ring sequences are enabled. 6. If (n) = 0 for some n 2 INN then M(m ) M (m ). Observe that the construction of ring sequences in point 2. uses minimal closed support T -invariants only. From point 5. we obtain that for all m 2 M(m ) the construction of the ring sequence such that m[ > m of point 2. remains valid. This proves that all m 2 M(m ) are recurrent states. 2 The result of Theorem 3.7 may seem obvious at rst sight: if all transitions are covered by T -invariants, then the Petri net has an invariant measure at the reachability set M(m ). As is shown in example 4.4 the result of Theorem 3.7 is not obvious. For the existence of an invariant measure (that is positive for all m 2 M(m )) additional assumptions on the Petri net are required. Assumption 3.2 provides a sucient condition for liveness. A result similar to the result of Theorem 3.7 is obtained by Whittle [23], p. 161, and by Boucherie and van Dijk [3] for reversible processes. In these references it is shown that the routing chain Y is reversible if and only if the Markov chain for the Petri net X is reversible (in the Markov chain de nition). For this result no additional assumptions (besides reversibility) are required on the T -invariants. The proof of Theorem 3.7 follows the same lines as the proofs in these references. An alternative formulation of Theorem 3.7 is presented in Coleman et al. [7]. In this reference assumptions are made on ring sequences from states back to a home state m . From Corollary 3.6 and Theorem 3.7 we obtain that a positive solution of the trac equations is sucient for X to be recurrent, that is for the existence of an invariant measure. Unlike the results for Jackson networks, the existence of a solution for the trac equations is not sucient for the invariant measure to be of product form. Additional conditions are required for this result. Below we present two results that can be found in the literature. The rst result is a standard product form result (Boucherie and van Dijk [4], Henderson and Taylor [13]). The marking independent solution of the trac equations is translated into a marking dependent solution with the same properties (Equation (3.4)). 1
1
2
2
ut
0
0
0
0
0
0
0
0
0
Theorem 3.8 Assume that an invariant measure y exists for the trac equations (3.2), and a function y : M(m ) ! IR such that for all n + I (t) 2 M(m ), t; s 2 T with p(I(t); I(s)) > 0, 0
+
0
12
4. Illustration of the results
y (n + I(t)) = y(I(t)) : y (n + I(s)) y(I(s))
(3.4)
Then m(m) = (m)y(m), m 2 M(m ), is an invariant measure of the Markov chain X describing the stochastic Petri net. If B ? = Pm2M m0 m(m) < 1, then (m) = B (m)y(m), m 2 M(m ), is an equilibrium distribution of X. 0
1
(
)
0
The function y guring in (3.4) is usually dicult to obtain from (3.4). The following theorem, taken from Coleman et al. [6], provides a product form solution for y under additional assumptions on the incidence matrix and y, the solution for the trac equations.
Theorem 3.9 Assume that y is an invariant measure for the trac equations. Then y satisfying (3.4) has the form y (m) =
N Y i=1
ci(y)m i
( )
if and only if Rank(A) = Rank([AjC(y)]); where [AjC(y)] is the matrix A augmented with the row C(y), de ned as C(y)j = log [y(I(tj ))=y(O(tj ))] ; j = 1; : : : ; M: In this case the N -vector c(y) = (log ci (y); i = 1; : : : ; N ) satis es the matrix equation c(y)A + C(y) = 0: 4. Illustration of the results
In this section we present some (motivating) examples illustrating the structural characterisation of stochastic Petri nets that allow for a positive solution of the trac equations. In particular, in section 4.1 we will relate our results to the results available in the literature. It will be shown that the sucient conditions available in the literature are too strong, and that the necessary conditions available are not sucient, although both conditions are very close to our necessary and sucient condition. In section 4.2 we will discuss Jackson networks. The main reason for the inclusion of this example is that necessary and sucient conditions for the existence of a positive solution for the trac equations are very elegant. Then, in section 4.3 we discuss batch routing queueing networks. It is shown that Assumption 3.2 basically comes down to the characterisation of `single batch' Markov chains underlying the process. Finally, in section 4.4 we will show that the assumption of closedness in Assumption 3.2 is essential. We will show that the assumption that all transitions are covered by minimal support T -invariants (without closedness) is not sucient for liveness of the Petri net.
13
4. Illustration of the results p1
p1
p2
t
1
t
t
t
t3
4
t1
3
2
p3
p4
p2 t4
t
5
Figure 2: a.
t2
Figure 2: b.
4.1 Relation with the literature
A necessary condition for the existence of a positive solution of the trac equations is derived by Henderson et al. [11], Corollary 1: for all g 2 R(T ) = St2T fI(t) [ O(t)g there exist t; s 2 T such that g = I(t), as well as g = O(s), that is R(T ) is a closed set. Comparison shows that Assumption 3.2 is more restrictive. In Assumption 3.2 the output bag of a transition t 2 kxk must be the input bag of the next transition in x, where kxk is required to be a closed set, whereas in the necessary condition of Henderson et al. kxk is not required to be a closed set. This is an important di erence, as is illustrated by the Petri net of Figure 2a, taken from Coleman [5], where the example is given to illustrate that the condition of Corollary 1 of Henderson et al. [11] is not sucient for the existence of a solution for the trac equations. Consider the Petri net of Figure 2a. The net has 3 minimal support T -invariants: x = (10010), x = (00101), x = (12001), of which x , and x have closed support, but x does not have closed support. Transition t is contained in kx k only. Therefore Assumption 3.2 is violated. The set of all transitions, T = ft ; t ; t ; t ; t g is a closed set. For transitions t ; t ; t ; t this is obvious since these transitions are contained in kx k [ kx k, and we have O(t ) = I(t ), I(t ) = O(t ). Therefore, the necessary condition of Henderson et al. [11], Corollary 1 is satis ed. It can easily be veri ed that the trac equations do not have a positive solution. A sucient condition for the existence of a positive solution of the trac equations is presented by Donatelli and Sereno [8]. There it is shown that the equilibrium distribution is of product form if all minimal support T -invariants are minimal closed support T -invariants. This implies the existence of a positive solution of the trac equations. In Assumption 3.2 we require only that some of the minimal support T -invariants have closed support and that the union of these closed supports equals the set of all transitions. This is an important di erence, as can be seen from the Petri net of Figure 2b, taken from Coleman et al. [6]. 1
2
3
1
3
2
3
2
1
1
2
1
3
4
5
2
1
2
5
2
3
4
5
14
4. Illustration of the results
Consider the Petri net depicted in Figure 2b. This Petri net has 4 minimal support T -invariants x = (1100), x = (0011), x = (2001), and x = (0210), of which x and x have closed support. Since kx k [ kx k = T , the Petri net satis es Assumption 3.2, and a positive solution exists for the trac equations. Note that the net does not satisfy the sucient condition of Donatelli and Sereno, since x and x do not have closed support. 1
2
2
3
1
4
1
2
3
4
4.2 Jackson networks
Consider an open or closed Jackson network consisting of N stations. The service requirement at station i is i, and is worked o at rate i(m(i)) when m(i) customers are present in the queue. If the network is open customers arrive to the network at Poisson rate , and de ne () 1. Customers leaving station i select station j with probability pij , j = 1; : : : ; N , and leave the network with probability pi . Customers arriving to the network select queue j with probability p j . The transition rates for the Jackson network have the form (3.1) 0
0
0
0
q(m; m ? ei + ej ) = i i(m(i))pij ; where transition tij corresponds to a customer moving from queue i to queue j (ek denotes the k-th unit vector). The trac equations are N X j =0
fciipij ? cj j pjig = 0; i = 1; : : : ; N:
(4.1)
The trac equations correspond to the global balance equations for the Markov chain representing the queueing network containing at most 1 customer, and ci is the equilibrium probability that this customer is at queue i. If this Markov chain has no transient states, then (4.1) as a set of equations for ci has a positive solution. From this observation we obtain that Assumption 3.2 characterises `single batch' Markov chains for the batch routing process. Observe that the existence of a positive solution of the trac equations (4.1) is necessary and sucient for the equilibrium distribution to be of product form: N Y (m) = B (m) ckm k : ( )
k=1
This result cannot be concluded for the general network of section 3 (cf. Theorems 3.8, 3.9). 4.3 Batch routing queueing networks
Batch routing queueing networks with a product form equilibrium distribution were introduced in [12], and are closely related to discrete-time queueing networks (see Boucherie and van Dijk [4]). The transition rates are
15
4. Illustration of the results
q(m; m ? g + g0) = (g) ( m(m?)g) p(g; g0); where g = (g(1); : : : ; g(N )) is the batch of customers leaving the queues, and g0 = (g0(1); : : : ; g0(N )) the batch of customers arriving at the queues. Here (g) (m ? g)= (m) is the state-dependent service-probability of servicing a batch g in state m, and p(g; g0) is the routing probability. The trac equations are X fc(g)(g)p(g; g0) ? c(g0)(g0)p(g0; g)g = 0; g0
a set of equations that is equivalent to the set of equations (3.3). A typical choice of the routing probabilities for closed networks is (Boucherie and van Dijk [4]) !Y N N Y X g ( i ) g 0 p p(g; g ) = 8 ij ; g ; : : : ; g 9i i iN j g ; i ; : : : ; N; j ; : : : ; N > > < Pg ; g p ; = > ; : P =1gg gg0 ij ;; ij ; ;: :: :: :; ;N;N > ij
ij
ij N
=1 0
ij
=
( )
=
( )
ij
j N
=1
ij
i
=1 = 0 if ij = 0
=1
:
1
=1
=1
=1
corresponding to the assumption that customers in the batch route independently according to the routing probabilities pij as de ned for Jackson networks. It can easily be shown that N cg k Y 1 c(g) = (g) gk(k)! k ( )
=1
is a solution of the trac equations if fcigNi is a positive solution of (4.1). =1
4.4 Liveness
When the equilibrium behaviour of stochastic Petri nets is of interest, a natural assumption is that all transitions are covered by minimal support T -invariants. For bounded nets this assumption is necessary for liveness (Silva [22]). If this assumption is not satis ed, then there exists a transition, say t , that is enabled in a reachable marking m, and t 62 Shi kxik (if t is never enabled, then we can delete t from T ). Let t re in marking m. Then there exists no ring sequence from m ? I(t ) + O(t ) back to m (otherwise t would be contained in a T -invariant). Thus m is a transient state and does not appear in the equilibrium description of the stochastic Petri net. As a consequence, both m and t can be deleted from the description of the Petri net. 0
0
0
=1
0
0
0
0
0
0
16
5. Conclusion
Figure 3: a.
Figure 3: b.
Figure 3: c.
As can be seen from the Petri net of Figure 3b, the assumption that all transitions are covered by T -invariants is necessary, but not sucient for liveness of the Petri net. For liveness additional assumptions are required. Assumption 3.2 guarantees liveness of the Petri net. As can be seen from Figure 3a, and 3c, Assumption 3.2 is sucient, but not necessary. Comparison of Figure 3b, and 3c, however, shows that the property of liveness is very complicated since Petri nets that are almost identical may show completely di erent behaviour. Therefore, a characterisation of liveness such as provided by Assumption 3.2 is of interest on its own. 5. Conclusion
This paper has introduced a new characterisation of the structure of batch routing queueing networks and stochastic Petri nets that is necessary and sucient for the existence of a positive solution for the trac equations. An interesting second result is that the structural characterisation is also sucient for the process to be recurrent (cannot have deadlocks). The structure of Petri nets satisfying the new characterisation stating that all transitions are covered by closed support T -invariants is investigated in several theoretical results and illustrated by examples. It is the new notion of minimal closed support T -invariants that has allowed us to start the analysis of the process directly from the incidence matrix of the net. This should be contrasted with results in which the analysis is performed based on the assumptions that a solution exists for the trac equations. In fact, this note has bridged the gap between the assumption usually made as a starting point for the analysis and the net-structure of the underlying Petri net. References
1. Ajmone Marsan, M., Balbo, G., Bobbio, A., Chiola, G., Conte, G. and Cumani, A.
References
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
17
(1989) The e ect of execution policies on the semantics and analysis of stochastic Petri nets, IEEE Transactions on Software Engineering 15, 832-846. Baskett, F., Chandy, K.M., Muntz, R.R. and Palacios, F.G. (1975) Open, closed and mixed networks of queues with di erent classes of customers, Journal of the ACM 22, 248-260. Boucherie, R.J. and Van Dijk, N.M. (1990) Spatial birth-death processes with multiple changes and applications to batch service networks and clustering processes, Advances in Applied Probability 22, 433-455. Boucherie, R.J. and Van Dijk, N.M. (1991) Product forms for queueing networks with state dependent multiple job transitions, Advances in Applied Probability 23, 152-187. Coleman, J.L. (1993) Stochastic Petri nets with product form equilibrium distributions. Ph.D. thesis, University of Adelaide. Coleman, J.L., Henderson, W. and Taylor, P.G. (1992) Product form equilibrium distributions and an algorithm for classes of batch movement queueing networks and stochastic Petri nets, Research Report, University of Adelaide. Coleman, J.L., Henderson, W., Pearce, C.E.M. and Taylor, P.G. (1993) A note on the correspondence between product-form batch-movement queueing networks and single-movement networks, Research Report, University of Adelaide. Donatelli, S. and Sereno, M. (1992) On the product form solution for stochastic Petri nets, Proceedings of the 13th international conference on application and theory of Petri nets, Sheeld, UK, 1992, 154-172. Frosch, D. (1992) Product form solutions for closed synchronized systems of stochastic sequential processes, Forschungsbericht Nr. 92-13, Universitat Trier, Mathematik/Informatik. Frosch, D. and Natarajan, K. (1992) Product form solutions for closed synchronized systems of stochastic sequential processes, Proceedings of 1992 International Computer Symposium, December 13-15, Taichung, Taiwan, 392-402. Henderson, W., Lucic, D. and Taylor, P.G. (1989) A net level performance analysis of stochastic Petri nets, Journal of the Australian Mathematical Society Series B 31, 176-187. Henderson, W., Pearce, C.E.M., Taylor, P.G. and Van Dijk, N.M. (1990) Closed queueing networks with batch services. Queueing Systems 6, 59-70. Henderson, W. and Taylor, P.G. (1990) Open networks of queues with batch arrivals and batch services, Queueing Systems 6, 71-88. Jackson, J.R. (1957) Networks of waiting lines, Operations Research 5, 518-521. Kelly, F.P. (1979) Reversibility and stochastic networks. Wiley.
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