Advances in Petri Nets 1985 (Lecture Notes in Computer Science 222), pp.478-498.
Copyright
c
1986 Springer–Verlag. The original publication is available at www.springerlink.com.
DOI 10.1007/BFb0016227.
M–TIMED PETRI NETS, PRIORITIES, PREEMPTIONS,
AND PERFORMANCE EVALUATION OF SYSTEMS
W.M. Zuberek
Department of Compute r Science
Memorial University of Newfoundland
St.John’s, Canada A1C 5S7
Abstract
In M –time d Petr i nets, firing times are exponentially distributed random variables associated
with transitions of a net. Several classes of M–timed Petri nets are discussed in this paper to
show increasing “modelling power ” of different nets. Conflict–free nets can model M and E
k
–type
queueing systems. Free—choice nets can also represent H
k
–type systems. Systems with several
classes of users and with service priorities assigned to user classes require net s with inhibitor arcs.
Preemption of service can be represented by extended nets with escape (or gener aliz ed inhibitor)
arcs. Finally, to provide flexible modelling of scheduling and decision strategies, enhanced Petri nets
are introduced with two classes of transitions, immediate and timed ones, and with (exponentially
distributed) firing times associated with the timed transitions. It is shown that the behaviour
of bounded M–timed Petri nets can be represented by fi ni te “state” graphs which are finite-state
continuous-time homogeneous Markov processes. Stationary probabilities of states can thus be
obtained by standard techniques used for analysis of Markov chains, and then operational analysis
can be applied for performance evaluation. Simple models of interactive sy s te ms are used as an
illustration of modelling.
1. INTRODUCTION
A Petri net [1,6,23] is an abstr act, formal model of sys t ems with interacting, concurr e nt or
parallel components. Petri nets have been successful ly used in modelling, validation and analys is of
systems in which it is possible for some events to occur concurrently but ther e are constraints on the
concurrence, precedenc e, or frequency of thes e occurrences [6,9,11,14,15,22]. Petri nets, however,
are not complete enough for the study of performance issues since no assumption is made on the
duration of systems activities. Several different concepts of “timed” Petri nets [5,8,19,24,25,26,27,28]
and s tochastic Petri nets [3,4,20,21] have been proposed by assigning (deterministic or stoch asti c)
firing and /or enabling times to the transitions and/or places of Petri nets.
The approach described in this paper is a continuation of the approach originated by Ramchan-
dani [24] and used to model the performance of digital systems at the regis te r transfer level [27]
and to study communication protocols [25] when fixed (or deterministic) firing times can be used.
In M–timed Petri nets, similarly as in stochastic nets [20,21], the firing times are exponentially dis-
tributed random variables, and their rates are assigned to transitions of a net. In inhibitor Petri nets,
inhibitor arcs [1,2,6,23] are used to represent priorities of simultaneous events. In extended Petri
nets, escape (or break) arcs are introduced to interrupt firing transitions and to model preemptions
of busy s er vers. In enhanced Petri nets [29,30], the transitions are partitioned into two classes, the
immediate and timed transitions, and the firing rates of exponentiall y distrib u ted firing times are as-
signed to timed transitions, as in generalized stochastic nets [3]. However, in ( bas ic and generalized)
stochastic nets the firings of transitions ar e instantaneous events , and the tokens actually remain in
the input places of firing transitions. The “state” space is thus conveniently determined by th e set
M-timed Petri nets, priorities, preemptions and performance evaluation of systems 479
of reachable markings, but multiple simultaneous events cannot be directly represented in such a
model, and this introduces some restrictions on modelling of even quite si mple systems [28]. More-
over, the stochastic approach cannot be applied to models in which the set of “states” (or discrete
behaviour of the model) is only loosely related to the set of reachable markings. In M–timed Petri
nets, the tokens are removed from corresponding places at the beginning of transition firings and
remain in transitions for the whole period of firing. The “state” d es cribes the distribution of tokens
in places as well as transitions, and this removes many restrictions of the stochastic approach. For
different classes of M–time d nets, a uniform discrete–s tate continu ous –time de s cr i pt ion is introduced
which represents the behaviour of nets by equivalent continuous–time homogeneous Markov chains
that can be generated directly from net specifications. For bounded nets, stationary probabilities
of system states can thus b e obtained by standard techniques, and this provides many performance
measures such as utilization of systems components, average queue lengths, average waiting times,
etc.
This paper is organized in 6 main sections. Section 2 contains definit ions of general conc ep ts for
conflict-free M–timed Petri nets . Free–choice M–timed Petri nets are discussed in section 3. Section
4 introduces inhibitor free–choice M–timed Petri nets, and section 5 extended free–choice M–timed
Petri nets. Enhanced free–choice M–timed Petri nets are described in section 6. Several simple
models of interactive computer systems are used as an illustration of performance evalu ation.
2. CONFLICT–FREE M–TIMED PETRI NETS
A basic Petri net N is a triple N = (P, T, A) where:
P is a finite, nonempty set of places,
T is a finite, nonempty set of transitions,
A is a set of directed arcs which connect places with transitions and tr ansi ti ons with places,
A ⊂ P × T ∪ T × P , such that:
∀(t ∈ T ) ∃(p
i
, p
j
∈ P ) (p
i
, t) ∈ A ∧ (t, p
j
) ∈ A.
A place p is an input (or an output) place of a transition t iff there exists an arc (p, t) (or (t, p),
respectively) in t he s et A. The sets of all inpu t and output places of a transition t are denoted by
Inp(t) and Out(t), respectively. Similarly, the sets of input and output tr an si tion s of a place p are
denoted by Inp(p) and Out(p). The n otation is extended in an obvious way to sets of places and
transitions, e.g., Inp(P
i
), Out(T
j
), et c.
A place p is shared iff it is an input place for more that one transition. A net is conflict-free if
it does not contain shared places.
A marked Petri net M is a pair M = (N, m
0
) where :
N is a Petri net, N = (P, T, A),
m
0
is an initial marking function which assigns a nonnegative nu mbe r of so called tokens to
each place of the net, m
0
: P → {0, 1, ...}.
Let any function m : P → {0, 1, ...} be called a marking in a net N.
A transition t is enabled by a marking m iff every input place p ∈ Inp(t) contains at least
one token, m( p) > 0. The set of transitions enabled by a marking m is denoted by T (m). Every
transition enabled by a marking m can fire. When a transition t fires, a token is removed from each
M-timed Petri nets, priorities, preemptions and performance evaluation of systems 480
of t’ s input places and a token is added to each of t’s output places. This determines a new marking
in a net, a new set of enabled transitions, and so on.
A marking m
j
is directly reachable from a marking m
i
in a net N iff there exists a transition t
enabled by the marking m
i
, t ∈ T (m
i
), s uch that:
∀(p ∈ P ) m
j
(p) =
m
i
(p) − 1, if p ∈ Inp(t) − Out(t),
m
i
(p) + 1, if p ∈ Out(t) − Inp(t),
m
i
(p), otherwise.
A firing sequence of a marking m is any sequence of transitions (t
j
1
, t
j
2
, ...) such that for k =
1, 2, ..., and for m
i
0
= m, t
j
k
∈ T (m
i
k−1
) and m
i
k
is dir e ct ly reachable from m
i
k−1
by firing t
j
k
.
A mark in g m
j
is reachable from a marking m
i
in a net M iff there exists a firing sequence which
transforms m
i
into m
j
, i.e., if there exists a sequence of marki ngs (m
i
0
, m
i
1
, m
i
2
, ..., m
i
n
) s u ch that
m
i
0
= m
i
, m
i
n
= m
j
, and each marking m
i
k
is directly reachable from the marking m
i
k−1
for
k = 1, ..., n.
A set M(M) of reachable markings of a Petri net M is the set of all mar ki ngs which are reachable
from th e initial marking m
0
.
A net M is bounded if there exists a positive integer k such that each marking in the set M(M)
assigns at most k tokens to each place of the net
∃(k > 0) ∀(m ∈ M (M)) ∀(p ∈ P ) m(p) ≤ k.
If a net M is bounded, its reachability set M(M) is finite. Only bounded Petri nets are considered
in this paper.
A marking graph G of a marked Petri net M is a labeled directed graph G(M) = (W, D, u) where:
W is a set of vertices which is equal to the set of reachable markings of the net M, W = M(M),
D is a s et of directed arcs, D ⊂ W × W , such that (m
i
, m
j
) is in D iff m
j
is directly reachable
from m
i
in M,
u is a labeling function which assign s a subset of the set of transitions to each arc (s
i
, s
j
) in the
set D, u : D → 2
T
, in such a way that u(m
i
, m
j
) c ontains all those transitions t
ij
∈ T (m
i
),
firing of which transforms m
i
into m
j
.
In timed Petri nets [24,25,26,27,28], each t r ans iti on t takes a “real” time to fire. When a transition
t is enabled, a firing can be initiated by remov in g tokens from t’s input places. The tokens remain
in the transition t for the “firing time”, and then the firing terminates by adding tokens to each of
t’s output places. Each of the firings is initiated in the same instant of time in which it is enabled.
If a transition is enabled while it fires, a new, independent firing can be initiated.
The firing times of transitions can b e described in several ways. In D–timed Petri nets [8, 24, 25,
26, 27] they are deterministic (or constant), i.e., there is a positive (rational) numb er assigned to
each transition of a net which determines the duration of transition’s firing. In M–timed Petri nets
[28,29] (or st ochastic Petri ne ts [3,4,20,21]), the firing times are exponentially distributed r and om
variables, and the corresponding firing rates are assigned to transitions of a net. The memoryless
property of the exponential distribution is the key factor in analysis of M–timed Petri nets.
A conflict- fr e e M–timed Petri net T is a pair T = (M, r) where:
M is a conflic t- fr ee bounded Petri net, M = (N, m
0
), and N = (P, T, A),
M-timed Petri nets, priorities, preemptions and performance evaluation of systems 481
r is a firing–rate function which assigns a positive real number r(t) to each t r ans iti on t of the net,
r : T → R
+
, and R
+
denotes the set of positive real numbers; the fir in g time of a tr an si tion t
is a random variable v(t) with t he distribution function
Prob(v(t) > x) = e
−x∗r(t)
, x > 0.
The memoryless property of exponential distributions means that if the duration v of a certain
activity (e.g., a firing time) is distributed exponentially with parameter r, and if that activity is
observed at time y after its beginning, then the remaining duration of the activ ity is independent of
y and is also distributed exponentially with parameter r:
Prob(v > y + x | v > y) = Prob(v > x) = e
−x∗r
.
The ex ponential distribution is the only continuou s distribution with the memoryless property.
A state s of an M –time d Petri net T is a pair s = (m, f) where:
m is a marking function, m : P → {0, 1, ...},
f is a firi ng function which indicates (for each transition of the net) the number of active firings,
i.e., the number of firings which have been initiated but are not yet terminated (i.e., are
“in pr ogre s s” in th e state s), f : T → {0, 1, ...}.
The in it ial state s
1
of a conflict-free net T is a pair s
1
= (m
1
, f
1
) where :
∀(t ∈ T ) f
1
(t) = min
p∈Inp(t)
(m
0
(p)),
∀(p ∈ P ) m
1
(p) = m
0
(p) −
X
t∈Out(p)
f
1
(t).
A state s
j
= (m
j
, f
j
) is directly t
k
–reachable from a state s
i
= (m
i
, f
i
) iff the following conditions
are satis fi ed :
(1) f
i
(t
k
) > 0,
(2) ∀ (p ∈ P ) m
ik
(p) = m
i
(p) +
1, if p ∈ Out(t
k
),
0, otherwise,
(3) ∀ (t ∈ T ) e
ℓ
(t) = min
p∈Inp(t)
(m
ik
(p)),
(4) ∀ (p ∈ P ) m
j
(p) = m
ik
(p) −
X
t∈Out(p)
e
ℓ
(t),
(5) ∀ (t ∈ T ) f
j
(t) = f
i
(t) + e
ℓ
(t) −
1, if t = t
k
,
0, otherwise.
The state s
j
which is directly t
k
–reachable from the state s
i
, is thus obtained by the termination
of a t
k
firing (1), updating the marking of a net (2), and then initiating new firings (if any) which
are det er min ed by the func ti on e
ℓ
(3, 4 and 5).
Similarly as for reachable markings, a state s
j
is re achable from a state s
i
if th er e is a sequence
of directly reachable states from the state s
i
to the state s
j
. Also, a set S(T) of reachable states is
defined as the set of all states of a net T which are reachable from the initial states. For bounded
conflict–free nets the sets of reachable states are finite.
A state graph G of an M–timed Petri net T is a labeled directed graph G(T) = (V, D, u) where:
M-timed Petri nets, priorities, preemptions and performance evaluation of systems 482
V is a set of vertices which is equal to the set of reachable states of the net T, V = S(T),
D is a set of directed arcs, D ⊂ V × V , such that (s
i
, s
j
) is in D iff s
j
is directly reachable
from s
i
in T,
u is a tr ans iti on–rat e function which assigns the rate of transitions from s
i
to s
j
to each arc
(s
i
, s
j
) in the set D, u : D → R
+
, in such a way that if s
j
is directly t
k
–reachable from
s
i
= (m
i
, f
i
), th en
u(s
i
, s
j
) = r(t
k
) ∗ f
i
(t
k
).
It sh oul d be noticed that state graphs of conflict–free bounded timed Petri nets are finite
continuous–time homogeneous Markov chains [13,16]. The stationary (or equilibrium) probabili-
ties x(s) of the states s ∈ S(T) are thus obtaine d from the state–transition rates by solving a system
of simultaneous line ar equations [12,16]
X
1≤j≤N
u(s
j
, s
i
) ∗ x(s
j
) = x(s
i
)
X
1≤j≤N
u(s
i
, s
j
); i = 1, ..., N − 1
X
1≤i≤N
x(s
i
) = 1
where N is the number of states in the set S(T).
Example. For a simple conflict–free M–timed Petri net T
1
shown in Fig.1a (as usual, places are
represented by circles, tr ans it ions by bars, the initial marking is represented by dots inside cir cl es ,
and the firing rate functions is given as an additional description of transitions ), the derivation of
the set S(T
1
) of reachable state s is shown in Tab.1 which also contains the stationary probabilities
of the states, x(s), s ∈ S(T
1
).
t3
1
p1
t1
5
t2
2
p4
2
5
7
4
1
3
6
(a)
(b)
p2
p3
Fig.1. Conflict-fr ee M–timed net T
1
(a) and its state graph (b).
The Petri net from Fig.1a is a model of a simple interactive computer system with a finite popu-
lation of users (or terminals — transition t
3
) and with a central server composed of two consecutive
stages (transitions t
1
and t
2
). The place p
2
models the queue of waiting jobs, and the place p
1
controls the number of server’s channels (the marking of p
1
models the number of idle channels
which can be equal to 0 or 1 in this example). The number of users (or terminals) in the sys te m