15-th ACM Annual Computer Science Conference (CSC’87); St. Louis, MO, 1987, February 17–19, 1987, pp.261–268.
Copyright
c
1987 ACM (DOI 10.1145/322917.322959).
MODIFIED M-TIMED PETRI NETS IN MODELLING
AND PERFORMANCE EVALUATION OF SYSTEMS
W.M. Zuberek
Department of Computer Science, Memorial University
St.John’s, NL, Canada A1B 3X5
Abstract
Modified M-timed Petri nets are Petri nets with ex-
ponenti ally distributed firing times and with generalized
inhibitor arcs t o interrupt firing transitions. It is shown
that the behavior of modified free-choice M-timed Petri
nets can be represented by probabilis t ic state graphs,
stationary probabilities of states can thus be obtained
by standard techniques used for analysis of continuous-
time homogeneous Markov chains. An immediate appli-
cation of such a model is performance analysis of queue-
ing systems with exponentially distributed service and
inte r arr i val times, and with priority and/or preemptive
sch ed uli ng disciplines. Simple models of computer sys-
tems with different scheduling strategies are us ed as an
illustration of modelling and performance analysis.
1. INTRODUCTION
Petr i nets [1,7,11] have been successfully used in mod-
elling, validation and analysis of systems of events in
which it is possible for some events to occur concur-
rently, but there are constraints on the occ ur r e nc e,
precedence, or frequency of these occurrences [6,11]. Ba-
sic Petri nets, however, are not complete enough for the
study of systems p e r for mance since no assumption is
made on the duration of systems events. Timed Petri
nets have been introduced by Ramchandani [12] by as-
signing ”firing times” to the transitions of Petri nets.
Sifakis [14] proposed another definition of a timed Petri
net by assigning ”enabling times” to places of a net.
Merlin and Farber [9] discussed timed Petri nets where
a time threshold and maximum delay wer e assigned to
each transition of a net to allow modelling of timeouts
used to recover from failur es in communication sy s tems .
Razouk [13] discussed yet another class of timed Petri
nets with enabling as well as firing times, and demon-
strated the derivation of (symbolic) performance expres-
sions for communication protocols. Another approach
to Petri nets ”with time” assumes that the firing times
are exponentially distributed random variables, and the
corresponding rates are assigned to transitions of a net;
such nets are called stochastic Petri nets [2,10]. In gen-
eralized stochastic Petri nets [2] the set of transitions is
subdivided into two classes of transitions, timed and im-
mediate ones. Timed transitions (as in stochastic nets)
have exponenti ally distributed firing times, while imme-
diate transitions fire in zero time, i.e., they are used
to represent logical conditions which do not contribute
to ”delay” time s. The memoryless property of the ex-
ponenti al distribution simplifies the ”state” description
and analysis of such nets , and, in fact, analysis of (basic
and generalized) stochastic nets is based on the sets of
reach able markings which are generated without timing
constraints. Consequently, the stochastic appr oach can
be used for only such models in which the state space is
isomorphic to the s pace of reachable markings [19], and
this may be a nontrivial task to check.
The formalism des cr i bed in this paper is a continua-
tion of the approach originated by Ramchandani [12,16]
and subsequently extended by multiple and inhibitor
arcs [17,18]. In (basic and extended) M-timed Petri nets,
however, firing transitions cannot be inte r r up te d, and
therefore preemptive disciplines are difficu lt to model
and analyze. In this paper, the basic Petri nets are en-
hanced by ”interrupt” arcs in ord er to suspend the pro-
cess of transition firi ng, as required in strict modelling of
preemptions. Similarly as in [2,10,17,18,19], firing rates
of exponentially distributed firing times are assigned to
transitions of a Petri net, and a ”state” description is
derived which represents the behavior of modifi ed free-
choice M-timed Petr i nets by continuous-time homo-
geneous Markov chains. The stationary probabilities
of the states can thus be obtained by standard tech-
niques, and this pr ovides many pe r for mance measures
such as utilization of systems components, average wait-
ing times and turnaround times or average throughput
rates, which can be derived automaticall y from model
specifications.
This paper is organized in 3 main sections. Section 2
contains definitions of basic concepts for modified free-
choice Petri nets. Modified M-timed Petri nets are in-
troduced in Section 3. Application of modified M-timed
Petr i nets to modelling and performance evaluation is
discussed in Section 4.
2. MODIFIED PETRI NETS
A modified Petr i net N is a quadruple N =
(P, T, A, C) where:
P is a finite, nonempty set of places,
Modified M-t imed Petri nets in modelling and performance evaluation of systems 262
T is a finite, nonempty set of transitions,
A is a nonempty set of directed arcs which connect
places with transitions and transitions with places
such that for e ach transition there is at least one
place connec te d with it
∀t ∈ T ∃p ∈ P : (p, t) ∈ A,
C is a (possi bl y empty) set of interrupt arcs which con-
nect places with transitions, C ⊂ P × T , and A and
C are disjoint sets.
A place p is an input (or an output) place of a transi-
tion t iff there exists an arc (p, t) (or ( t, p), respectively)
in the set A. The sets of all input and output places of
a transition t are denoted by Inp(t) and Out(t) , respec -
tive ly. S imi larl y, the sets of input and output transitions
of a place p are denoted by Inp(p) and Out(p). Also,
a place p is an interrupting place of a transition t iff
(p, t) ∈ C. The set of all interrupting places of t is de-
noted by Int(t), and the set of transitions connected
by interrupt arcs with a place p is denoted by Int(p),
Int(p) = {t|p ∈ Int(t)}. The notation is extended on
sets of places and transitions in the usual way.
A modified net N = (P, T, A, C) is singular iff the
input sets of transi ti ons with nonempty interrupting sets
are disjoint with interr up ti ng sets of other transitions
∀t ∈ T : Int(t) = ∅ ∨ Int(Inp(t)) = ∅
where ∅ denotes the empty set. The description of
non-singular nets must take into account ”sequences”
of consecutive interrupts in a net when one interrupted
transition, through its input places, interrupts another
transition(s). In singular nets t he r e is no such ”propa-
gation” of inte r r u pt s.
A marked Petri net M is a pair M = (N, m
0
) where:
N is a modified Petri net, N = (P, T, A, C),
m
0
is an initial marking func tion which assigns a non-
negative integer number of so called tokens t o each
place of the net, m
0
: P → {0, 1, ...}.
Let any function m : P → {0, 1, ...} be called a mark-
ing in a net N = ( P, T, A, C).
A transition t is enabled by a marking m iff every
input place of this transition contains at least one token
and every interrupting place of t contains zero tokens.
For ordinary nets (i.e., nets without time) the interrupt
arcs are thus equivalent to inhibitor arcs [1,11,17].
The set of all transitions enabled by a marking m is
denoted by En(m).
A place p is shared iff it is an inpu t place for more
than one transition. A shared place p is guarded iff for
each two different transitions t
i
and t
j
sharing p there
is another place p
k
such that p
k
is in the input set of
one of these t r ans iti ons and in the interrupt in g set of
the other one
∀t
i
∈ Out(p) ∀t
j
∈ Out(p) − {t
i
} ∃p
k
∈ P − {p} :
((p
k
, t
i
) ∈ A ∧ (p
k
, t
j
) ∈ C) ∨ ((p
k
, t
j
) ∈
A ∧ (p
k
, t
i
) ∈ C),
i.e., no two transitions from the set Out(p) can be en-
abled by the same marking m. A net is conflict-free iff
all its shared p lace s are guarde d.
A shared place p is free-choice (or extended free-choice
[7]) iff the input sets and i nterrupting sets of all transi-
tions shar in g p are ident ic al, i.e., iff:
∀t
i
, t
j
∈ Out(p)) : Inp(t−i) = Inp(t
j
)∧Int(t
i
) = Int(t
j
).
A net is free-choice iff all i ts s h are d place s are either
guarded or free-choice. Only free-choice Petri nets are
considered in this paper.
Ever y transition enabled by a marking m can fire.
When a transition fires, a token is r emoved from each
of its input places (but not from interr u pt in g places),
and a token is added to each of its output places. This
determines a new marking in a net, a new set of en able d
transitions, and so on.
A marking m
j
is directly reachable (or t
k
-reachable)
from a marking m
i
in a net N iff there exists a transition
t
k
enabled by the marking m
i
, t ∈ En(m
i
), such 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 marking m
j
is (generally) reachable from a marking
m
i
in a net N if ther e exists a sequence of markings
(m
i
0
m
i
1
m
i
2
...m
i
k
) such that m
i
0
= m
i
, m
i
k
= m
j
, and
each marking m
i
ℓ
is directly reachable from the marking
m
i
ℓ−1
for ℓ = 1, ..., k.
A set M(M) of reachable markings of a marked Petri
net M = (N, m
0
) is the set of all markings whi ch are
reach able from the initial marking m
0
(including m
0
).
3. MODIFIED M-TIMED PETRI NETS
In timed Petri nets , each transition t takes a ”real”
time to fir e , and therefore it is convenient to distinguish
three phases of transition firings, its initiation, delay,
and termination. When a transition t is enabled, a fir-
ing can be initiated by removing a token from each of
t’s input places. Th is token remains in the transition t
for the ”firing time”, and then the firing ter min ates by
adding a token to each of t’s outpu t places. Each of the
firings is initiated in the same instant of time in which it
becomes enabled. If a transit ion be comes e nab led wh ile
it fires, a new, indepe nd ent firing can be initiated. If
a net contains conflic ts , and there are several different
possibilities of firing transitions for the same marking,
Modified M-t imed Petri nets in modelling and performance evaluation of systems 263
the choice of actual transitions is assumed to be a ran-
dom pr oces s which can be described by corresponding
probabilities or probability distribution functions. In
modified timed Petri nets, a firing of a transition may
be interrupted i f the set of transition interr u pt in g places
becomes nonempty. If, d uring a firing period of such
transition t, all t’s interruptin g places become marked,
the firing of t ceases and the tokens removed from t’s
input places at the beginning of firing, are ”returned”
to their original places. It should be noticed that the
inte r r up ti ng places of t must be empty to initiate a fir-
ing of t; an interrupt can thus occur only as a result of
termination of another firing (or firings).
Since in timed nets all transition firings are initiated
in the same instants of ti me in which the transitions
become enabled, it is convenient to associate with each
marking m a set of all possibilities of new firings (in
nets with conflicts there are usually several such possi-
bilities). The set of selection functions describes all such
possibilities.
A selection function e of a marking m in a net N is
any function e : T → {0, 1, ...} such that
(1) there exists a sequence of transitions u =
(t
i
1
, t
i
2
, ..., t
k
) in which t
i
j
∈ En(m
i
j−1
) for j =
1, ..., k, and for m
i
0
= m where
∀p ∈ P : m
i
j
(p) = m
i
j−1
(p)−
1, if p ∈ Inp(t
i
j
),
0, otherwise,
(2) the set of transitions enabled by m
i
k
, En(m
i
k
), is
empty,
(3) for each tinT , e(t) is equal to the number of occur-
rences of t in the sequen ce u;
i.e., a selection function e indicates (by nonzero values)
all those transitions which can initiate their firings si-
multane ous ly (and some transitions may initiate several
firings). The set of all selection functi ons of a marking
m is denoted by Sel(m).
A marked net M is simple if all selection f un ct ions of
all reachable markings assign at most one firing to each
transition of M, i.e.
∀m ∈ M(M) ∀e ∈ Sel(m)) ∀t ∈ T : e(t) ≤ 1.
Only simple nets are discussed in this paper. Non-
simple nets can be described by a straightforward ex-
tension of simple nets [ 18,19] that takes into account
combination s of multiple firings and their corresponding
probabilities. It should be noticed that in M-timed nets
without multiple arcs, non-simple behaviour can occur
only in the initial states, when the initial marking func-
tion introduce s ”multiple enablings” of some transitions.
Moreover, such initial markings can usually be converted
to equivalent ones, which preserve the behaviour of a
net, and which provide simple selection fu nc tion s.
An M-timed modified free-choice Petri net T is a
triple T = (M, c, r) where:
M is a modified free-choice marked Pe tr i net, M =
(N, m
0
), N = (P, T, A, C),
c is a choice function which assigns a free -choice prob-
ability to e ach transition t of the net in such a way
that for each free-choice place p:
X
t∈Out(p)
c(t) = 1,
and for all remaining transitions c(t) = 1,
r is a firing rate function which assigns a positive
real number r(t) to each transition t of the net,
r : T − > tarrowR
+
, and R
+
denotes the set of
positive real numbers; the firing time of a transi-
tion t is a random variable v(t) with the distribution
function
Prob(v(t) > x) = e
−x∗r(t)
, x > 0.
The memoryless (or Markov) prop er ty of the (nega-
tive ) exponential distribution is t he basic factor in de-
scriptions of M-timed Petri nets (the exponential dis-
tribution is the only continuous distr i bu tion with the
memoryless prope r ty). It means that if the duration v
of a certain activity (e.g., the firing time) is distributed
exponentially with parameter r, and if that activity is
observed at time y after its beginning, then the remain-
ing duration of the activity is independe nt of y and is
also distr ib ut ed exponentially with parameter r:
Prob(v > y + x|v > y) = Prob(v > x) = e
−r∗x
.
Consequently, the state descriptions of M-timed nets
represent only the actual configuration of a net, and
completely igmores t he history, or the sequence of pre-
vious state s .
The behavior of an M-timed Petri net can be de-
scribed by a set of states (or configurations of a net),
with state transitions corr es pond in g to termin ations of
firings (and initiations of new firings).
A state ss@ of an M-timed Petri net T is a pair of
functions s = (m, f) where:
m is a marking function, m : P → {0, 1, ...},
f is a firing function which indicates (for each transi-
tion of the net) the number of active firings, i.e.,
the number of firings which have been initiated but
are not yet terminated, f : T rightarrow{0, 1, ...}.
Modified M-t imed Petri nets in modelling and performance evaluation of systems 264
An initial state s of a net T is a pair s = (m) i, f
i
)
where f
i
is a s ele ct ion function from the set Sel(m
0
),
and the marking m
i
is defined by
∀p ∈ P : m
i
(p) = m
0
(p) −
X
t∈Out(p)
f
i
(t).
A free-choice M-timed net T may have several differ-
ent initial states.
A state s
j
= (m
j
, f
j
) is directly reachable (or (t
k
, e
ℓ
)-
reach able ) from t he state s
i
= (m
i
, f
i
) iff:
1. f
i
(t
k
) > 0,
2. e
ℓ
∈ Sel(m
ikj
),
3. ∀p ∈ P ) : m
j
(p) = m
ikj
(p) −
P
t∈Out(p)
e
ℓ
(t),
4. ∀t ∈ T : f
j
(t) = f
ik
(t) − d
ik
(t) + e
ℓ
(t),
where
5. ∀p ∈ P : m
ikj
(p) = m
ik
(p) +
P
t∈Out(p)
d
ik
(t),
6. ∀t ∈ T : d
ik
(t) = min(f
ik
(t), min
p∈Int(t)
(m
ik
(p)).
7. ∀p ∈ P : m
ik
(p) = m
i
(p) +
1, if p ∈ Out(t
k
),
0, otherwise,
8. ∀t ∈ T : f
ik
(t) = f
i
(t) −
1, if t = t
k
,
0, otherwise.
The state s
j
which is (t
k
, e
ℓ
)-reachable from the state
s
i
is thus obt aine d by the termination of a t
k
firing
(1), updating the marking and firing functions (7,8),
performing all the interrupts (if any) of firing transi-
tions which correspond to the updated marking, mik (5
and 6), and then initiating new firings (if any) which
are determined by a selection function e
ℓ
from the set
Sel(m
ikj
) (2,3 and 4). It should be noticed that the for-
mula (6) is a direct consequence of singularity and does
not take ”propagation” of interrupts into account. For
nonsingular nets, the description of state reachability
becomes a little mor e convoluted .
Similarly as for marked nets, a state s
j
is (generally)
reach able from a state s
i
if there is a sequence of directly
reach able states from the state s
i
to the state s
j
. Also,
the set S(T) of reachable states is defined as the set of
all states of a net T which are reach able from the initial
states of the net T (incl ud in g the initial state(s)).
A state graph G of an M-timed Petri net T is a labeled
directed graph G(T) = (V, D, u) where:
V is a set of vertices which is equal to the set of reach-
able 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 direct ly reachable from s
i
,
u is a transition-rate function which assigns the rate of
transition 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
(t
k
, e
ℓ
)-reachable fr om s
i
and s
i
= (m
i
, f
i
), then
u(s
i
, s
j
) = r(t
k
) ∗ f
i
(t
k
) ∗
Y
t∈T
c(t)
e
ℓ
(t)
.
It should be observed that the state graph of a mod-
ified free-choice M-timed Petri net is a continuous-time
homogeneous Markov chain. The stationary probabil-
ities x(s) of the states s ∈ S(T) can be obtained by
solving a system of simultaneous linear equations [5,8]
X
(s
j
,s
i
)∈D
u(s
j
, s
i
) ∗ x(s
j
) = x(s
i
) ∗
X
(s
i
,s
j
)∈D
u(s
i
, s
j
);
for i = 1, ..., K − 1
X
1≤i≤K
x(s
i
) = 1
where K is the number of states in the set S(T).
Example. Th e M-timed Petri net shown in Fig.1a (as
usual, places are represented by circles, transitions by
bars, interru pt arcs by small dots instead of arrowheads,
the initial marking by dots inside places, and the firing
rate function and the choice function are given as an
additional descriptions of transiti ons ) net contains one
guarded place, p
1
. The net is singular since it contains
only one transition (t
3
) with nonempty interrupting set.
The choice function, c, assigns the probability 1 to all
transitions since this net does not c ontain free-choice
places.
The state graph G(T
1
) is shown in Fig.1b, and the
derivation of the set of reachable states S(T
1
) is given
in Tab.1 which also contains the stationary probabilities
x(s
i
) of the states s
i
∈ S(T
1
). .
Fig.1. M-timed net T
1
(a) and its state graph ( b) .
Modified M-t imed Petri nets in modelling and performance evaluation of systems 265
m
i
f
i
d
ik
m
ikj
e
ℓ
s
i
x(s
i
) 1 2 3 4 5 1 2 3 4 t
k
1 2 3 4 1 2 3 4 5 1 2 3 4 s
j
u(s
i
, s
j
)
1 0.184 0 0 0 2 0 1 1 0 0 1 0 0 0 0 1 0 1 2 0 0 1 1 0 0 2 5.0
2 0 0 0 0 0 1 0 2 0 0 0 0 0 3 2.0
2 0.184 0 0 0 1 0 0 2 1 0 2 0 0 1 0 1 1 0 2 0 1 0 0 0 1 4.0
3 0 0 0 0 1 0 0 1 1 0 0 1 1 4 2.0
3 0.084 0 1 0 2 0 1 0 0 0 1 0 0 0 0 1 1 1 2 0 1 1 0 0 1 5.0
4 0.180 0 0 0 0 0 0 2 1 1 2 0 0 1 0 1 1 0 1 0 1 0 0 0 5 4.0
3 0 0 0 0 1 0 0 0 1 0 0 0 1 6 2.0
4 0 0 0 0 0 0 0 1 0 0 0 0 0 2 1.0
5 0.136 0 0 0 1 0 1 1 0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 4 5.0
2 0 0 0 0 0 1 0 1 0 0 0 0 0 7 2.0
4 0 0 0 0 0 0 0 2 0 0 0 0 0 1 1.0
6 0.107 1 0 0 0 0 0 2 0 2 2 0 0 0 0 1 1 0 0 0 1 0 0 0 8 4.0
4 0 0 0 0 1 0 0 1 0 0 0 1 0 4 2.0
7 0.051 0 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 5 5.0
4 0 0 0 0 0 1 0 2 0 0 0 0 0 3 1.0
8 0.057 0 0 0 0 0 1 1 0 2 1 0 0 0 0 1 0 1 0 0 0 1 0 0 6 5.0
2 0 0 0 0 0 1 0 0 0 0 0 0 0 9 2.0
4 0 0 0 0 0 0 0 1 0 0 0 0 0 5 2.0
9 0.016 0 1 0 0 0 1 0 0 2 1 0 0 0 0 1 1 1 0 0 1 1 0 0 8 5.0
4 0 0 0 0 0 1 0 1 0 0 0 0 0 7 2.0
Tab.1. The set of reachable states for T
1
.
4. PERFORMANCE EVALUATION
A very simple closed-network model of an interactive
system with 2 class e s of users (and j obs ) and with a p r i-
ority preemptive s cheduling is shown in Fig.2a. It con-
tains one ce ntral server P
c
with two queues of waiting
jobs, Q
1
and Q
2
(for class-1 and class-2 jobs, respec-
tive ly) , and n
1
(active) users in class-1 and n
2
(active)
users in class-2. All class-1 jobs have higher priority
than the class-2 ones, i.e., the class-1 jobs rece ive the
service before the jobs from class-2.
Fig.2. Closed network model os an interactive systems (a)
and its transition-rate diagram (b).
Suppose that the jobs of the same class are s tati st i-
cally i de ntical, that they are served by the First-Come-
First-Served discipline, and that the service times as well
as the user terminal (or ”thinkin g”) times are exponen-
tially distributed. Under these assumptions the number
of jobs in th e system (i.e., in the server and its waiting
queues) is a finite continuous-time homogeneous Markov
chain [5,8]. For n
1
= n
2
= 2, there are 9 states of this
Markov chain:
0: no jobs in the system;
1: a class-1 job in P
c
, empty queues;
2: a class-2 job in P
c
, empty queues;
3: a class-1 job in P
c
, 1 job in the queue Q
1
;
4: a class-1 job in P
c
, 1 job in the queue Q
2
;
5: a class-2 job in P
c
, 1 job in the queue Q
2
;
6: a class-1 job in P
c
, 1 job in Q
1
, 1 job in Q
2
;
7: a class-1 job in P
c
, 2 jobs in the queue Q
2
;
8: a class-1 job in P
c
, 1 job in Q
1
, 2 jobs in Q
2
.
The corresponding transition-rate diagram [6] is
shown in Fig.2b where a
1
, a
2
, d
1
and d
2
denote the
terminal rates for class-1 and class-2, and the service
rates for class-1 and class-2 j obs , respectively.
The same system can be modeled by the M-timed
Petr i net sh own in Fig.1a. The transitions t
1
and t
3
correspond to the central server processing class-1 jobs
(t
1
) and class-2 jobs (t
3
) with the service rates (or the
firing rates) equal to 5 and 2, respectively. The places
p
2
and p
4
model the waiting queues (Q
1
and Q
2
, re-
spectively). The transitions t
2
and t
4
correspond to the