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DOI: 10.15514/ISPRAS-2022-34(5)-12
Overapproximation of the Number of Active Timers in Timed-Arc Petri Nets Using DP-Systems
L.W. Dworzanski, ORCID: 0000-0002-0074- 7660 <leo@mathtech.ru> National Research University Higher School of Economics, Myasnitskaya st., 20, Moscow, 101000, Russia
Abstract. Timed-arcs Petri nets are a time extension of Petri nets that allows assigning clocks to tokens. System of dynamic points on a metric graph (DP-systems) is another dynamical model that is studied in discrete geometry dynamics; DP-system combines continuous time and discrete branching events and used, for example, in study of localized Gaussian wave packets scattering on thin structures. In recent works, asymptotic estimates of the growth of the number of points in dynamic systems on metric graphs were obtained. In this paper, we provide a mean to overapproximate the number of different values of timers for a subclass of timed-arc Petri nets by constructing a system of dynamic points on a metric graph and prove overapproximation of the number of timer values by the number of points in the system of dynamic points.
Keywords: metric graphs; timed-arc Petri nets; dynamic properties
For citation: Dworzanski L.W. Overapproximation of the Number of Active Timers in Timed-Arc Petri Nets Using DP-Systems. Trudy ISP RAN/Proc. ISP RAS, vol. 34, issue 5, 2022. pp. 183-194. DOI: 10.15514/ISPRAS-2022-34(5)-12
Acknowledgements. The reported study was funded by RFBR according to the research project 20-07-01103a.
Оценка сверху числа активных таймеров в сетях Петри с временными дугами с помощью динамических систем точек на
графах
Л.В. Дворянский, ORCID: 0000-0002-0074-7660 <leo@mathtech.ru> Национальный исследовательский университет «Высшая школа экономики», 101000, Россия, Москва, Мясницкая ул., 20
Аннотация. Сети Петри с временными дугами - это временное расширение сетей Петри (TaPN-сети), которое позволяет присваивать таймеры фишкам. Система динамических точек на метрическом графе (DP-система) это другая динамическая модель, которая рассматривается в теории геометрических дискретных динамических систем и, исторически, ее изучение мотивировано изучением распространения локализованных гауссовых волновых пакетов по тонким структурам. DP-система моделирует дискретные ветвящиеся события происходящие в реальном времени. В недавних работах были получены асимптотические оценки на рост числа точек в DP-системах на метрических графах. В данной работе мы предлагаем методы оценки сверху числа различных значений таймеров для подкласса сетей Петри с временными дугами с помощью построения DP-системы на метрическом графе по сети Петри и показывает, что количество различных значений таймеров в исходной сети Петри не превосходят количество точек в DP-системе. Это позволяет переносить известные оценки для DP-систем на сети Петри с временными дугами для выделенного подкласса.
Ключевые слова: метрические графы; сети Петри с временными дугами; динамические свойства
Для цитирования: Дворянский Л.В. Оценка сверху числа активных таймеров в сетях Петри с временными дугами с помощью динамических систем точек на графах. Труды ИСП РАН, том 34, вып. 5, 2022 г., стр. 183-194. DOI: 10.15514/ISPRAS-2022-34(5)-12
Благодарности: Работа выполнена при поддержке гранта РФФИ 20-07-01103 a.
1. Introduction
Petri nets are widely-used to model the behaviour of distributed concurrent computer systems and concurrent processes in biology, chemistry, physics, and other fields [1]. There are many time extensions of Petri nets were suggested [2, 3]. Simultaneously, almost any of semantical extensions makes Petri nets Turing-complete and many of general behavioral problems immediately become undecidable by Rice-Uspensky theorem. The widely known time extensions of Petri nets - Time Petri nets and Timed (Duration) Petri nets - are Turing-complete as they admit urgency and allow to model unbounded counters. Time extensions of Petri nets, as well as other real-time models, are under active study as, for many real-world software/hardware systems, time related aspects like performance, time-outs, delays, and latency are crucial for correct functioning [4-6]. A time semantics with restricted urgency was recently suggested for TaPN-nets in [7]; the suggested semantics allows urgent transitions to consume tokens only from the bounded places of a Petri net, and this restriction makes some behavioral problems decidable for TaPN-nets. Timed-arc Petri nets (TaPN-nets) are an extension of Petri nets with real-time semantics: tokens are assigned clocks [8]; the inscription on an incoming arc of a transition define tokens of which age can be consumed by the firing of the transition.
TaPN-nets could be used to model mobile agents moving among nodes or data packets being transmitted between nodes of a telecommunication system. The number of tokens with different timers corresponds to the number of different data packets in a system. Examples of systems where the number of different packets is an important characteristic - networks of self-driven cars, high-loaded telecommunication systems experiencing DDoS attack, underwater network of drone swarm with intensive communication. Excessive number of packets stipulates communication quality degradation or failure. Therefore, to assess the quality of system functioning, it is important to estimate the number of tokens-packets with different timer values. In this paper, we suggest to an approach to approximate the number of different timers in a TaPN by the number of dynamic points on metric graphs.
Metric graph is a graph with lengths assigned to edges. Some results towards dynamical characteristics of systems of dynamic points flowing along undirected edges of a metric graphs in both directions were recently obtained [9-13]. In such systems, when a point reaches a vertex of the graph, new points start moving along all the edges incident to the vertex. The growth of the number of points moving along edges and its asymptotics were studied in [11, 12] and, for some special cases, in [14].
The orientation of metric graph edges allows us to approximate TaPN-nets more precisely; thus, in [15], the asymptotics of points number growth were obtained for systems of dynamic points on directed Sperner graphs. And later in [16], it was extended to directed Hamiltonian graphs. The suggested simulation of a TaPN-net with a metric graph enables us to overapproximate the number of different clock values in the TaPN-net. A point in a constructed metric graph represents a clock value in the TaPN-net, and the event of the arrival of a point to a vertex simulates that the clock has reached a specific value. This allows us to estimate the number of different time events in a system modelled using TaPN-net. Translation to less expressive but more amenable to behavioural analysis or better studied models can be found in coverability analysis of Petri nets using Karp-Miller trees or in performance analysis of continuous timed Petri nets using Markov processes [17].
точек на графах. Труды ИСП РАН, том 34, вып. 5, 2022 г., стр. 183-194
Fig. i. TaPN-net model of an underwater wireless sensor network The frequency of time events and its growth can be crucial as it determines energy or traffic load of the system. The growth of frequency can be important for such systems as software defined networks when resource management is adaptive [18] or when communication is complicated due to media. As a motivational example, we can consider a TaPN-net model of an underwater wireless sensor networks [19, 20]. The communication among underwater sensors is complicated due to the transmission medium (salt water); energy consumption and packets collisions impede efficient implementation of acoustic underwater transmission technology. Consider TaPN-net model TaPNuwa of an underwater acoustic sensor network (UWA-SN) in Figure 1. Places sensorsi-3 model spatial locations of underwater sensors, which measure temperature and pH levels. Sensors are regularly issued to places sensorso and sensors2 from pools pooli and pooh. Each 30 time units, a sensor flows to another spatial position. As communication is complicated and energy is an important resource, the active participation of sensors in communication shall be reduced to minimum. Thus, sensors receive commands to probe the water from controllers controlleri and controller2. Signal emission of controllers is modelled with places tasksi-3. If a sensor is ready and receives a signal tasksi-3, then it takes takei-3 an assignment, processes processingi - 3 it, and returns backi-3 to waiting states sensorsi-3.
A collision in a node may occur when several signals reach the node simultaneously. Such collisions depend on the spatial position of nodes in UWA-SN. When such a collision occurs, the node signals emitters about the problem, and the emitters cooperate to avoid collisions. Thus, the appearance of a new time event can result in additional traffic in the network. When many collisions occur, such reconfiguration can impede the normal functioning; therefore, the growth rate of the number of points could be an important parameter of the systems [19, 20].
To overapproximate the growth of different time event in the system TaPN , we build a metric graph r. The growth rate of points in r is not lower than thus of the number of different token-clock values in the TaPN-net. Thus, if the growth rate of r is acceptable, then so is the growth rate in TaPN. Both models embrace real-time dynamics of discrete entities moving within a structure defined by a graph. We establish correspondence between a subclass of TaPN-nets and DP-systems. In Section
2, the notions of dynamical systems of points on metric graphs and timed-arc Petri nets are given. In Section 3, we provide the notion of overapproximation and supporting translation from a TaPN-net to a metric graph. In section 4, we provide a translation of a DP-system to a TaPN-net. Section 5 concludes the paper with some discussion on further research.
2. Preliminaries
By N, Q>o , R, we denote the sets of natural, non-negative rational, and real numbers, respectively. The set of open and closed intervals over R >o и is denoted by I (R >0). For a set S, a bag (multiset) m over S is a mapping m : S ^ N. The set of all bags over S is denoted by NS. We denote addition and subtraction of two bags by + and -, the number of all elements in m taking into account the multiplicity by |m|, and pointwise comparisons of bags by =, <, >, <, >, that are defined as miRm2=VsES:mi(s)Rm2(s) where R is one of =, <, >, < > We overload the set notation writing 0 for the empty bag and 6 for the element inclusion.
2.1. DP-systems on directed metric graphs
A directed metric graph Г is a graph consisting of set of vertices V, set of directed edges (arcs) E, and length function l mapping each arc a = <vi, V2>6 E to a positive real, i.e., l: E^ R+ [21]. The arc opposite to arc a = <v, v> is denoted by a, i.e., a = <Vj, v>. For two points x andy on the graph, metric p(x, y) is the shortest distance between them, where distance is measured along the arcs of the graph additively. A walk is a finite or infinite sequence of arcs which joins a sequence of vertices.
Definition 1 (System of dynamic points). A system of dynamic points (DP-system) on a metric graph Pr is a pair <r, P>, where r is metric graph and P is a set of points distributed on vertices and edges of T.
Dynamics of a DP-system Pr is defined as following. In the initial state, a set of points is distributed on the arcs and vertices of r. The position of point p on arc a is denoted by xa(p) or just x(p). When time starts to flow, each point p moves along its arc a direction. Each pointp located at vertex v, for each outgoing arc a incident to v, produces a new point po on each a, and p disappears (intuitively, this corresponds to wave packet scattering); each produced point po starts moving along corresponding a. All points move with the same constant velocity; and, due to new points generation, some arcs may carry more than one point. When a moving point reaches the end incident to vertex vo, again, on each outgoing arc incident to vo, a new point is generated. When more than one points reach a vertex simultaneously at t, on each outgoing arc, only one point is produced, as if only one point has reached the vertex at t; i.e., points met on a vertex fuse, and each coordinate of an arc can carry only one dynamic point.
In Figure 2, the initial set of points consists of two points in vertices vi and v3. The point at vi produces a new point on edge <vi, v2>. The point at v3 produces points on edges leading to vertices
О
О
■"4
Fig. 2. System of dynamic points Pr on metric graph Г
v2, V4, V5, V6, V7. After a time unit, there are no points in vi and v3 (coloured grey), but there are points (coloured black) moving from vi and v3 to their adjacent vertices.
The number of dynamic points in Pr at time t is denoted by Npr(t). The number of points on edge e at time t is denoted by Ne(t). For dynamical systems of points Pr and P 'r, we say that the growth rate of Pr is equal or less than thus of P'r■ if Vt 6 R+\Coll: Npr(t) < Npr(t), where Coll is the (countable) set of time points when more than one dynamic point meet on a vertex; we exclude such time points as technically the number of dynamic points decreases for these moments. In what follows, we discuss number of points implicitly omitting vertices collision time points.
2.2. Timed-Arc Petri nets
Petri nets is a classical well-known formalism for concurrent systems modelling. A place/transition net (PT-net) is a Petri net with black tokens, that are indistinguishable from each other. Definition 2 (Place/transition nets). A PT-net is a tuple <P, T,F,y>, where
• P and T are disjoint finite sets of places, respectively, transitions;
• F £(P x T) U (T x P) is a set of arcs (flow relation);
• y : F ^ N is a weight function.
For an element x E P u T, an arc <y,x> is called an input arc, and arc <x,y> - an output arc. Preset •xand a postsetx'are subsets of P u Tsuch that 'x = {y\<y,x> E F} and x' = {y\<x,y> E F}. A marking of N is a function m: P ^ N. A pair <N,m> of a PT-net and a marking is called a marked net. Let N = <P, T,F,y> be a PT-net. A transition t E T is enabled in a marking m iff Vp E 't ^ m(p) > y <p,t>. An enabled transition t can fire yielding a new marking m'(p) = m(p)-y <p,t>+y <t,p> for each p E P (denoted m ^t m' ). The set of all markings reachable from a marking m is denoted by R(m).
Now, we provide the definition of Timed-Arc Petri nets (TaPN-nets) with token-based time semantics and urgency [6, 8, 22].
Definition 3 (Timed-Arc Petri net with Urgency). A TaPN-net is a tuple TaPN = <N,y*,U>, where
• N = <P, T,F,y> is a PT-net called the skeleton of TaPN and denoted by S(TaPN);
• yt: P x T ^ I(R+ > 0) is a set of token-age constraints on arcs;
• U: T ^ Q>0 is a set of urgency constraints on transitions.
The marking m = <ms,mt,mu> of a TaPN-net TaPN consists of a marking ms of S(TaPN), a time marking mt : Tok(ms) ^ R>o that assigns clocks to tokens, and an urgency marking mU : T(ms) ^ R>o that assigns clocks to transitions, where T(ms) comprises all enabled transitions and Tok(ms) comprises all tokens of the marked PT-net <S(TaPN),ms>. The urgency constraint U(t) means that t must fire if t has been enabled for U(t) units of time. The token-age constraint yt(p,t) defines that a firing of t may consume only token z in p with mt(z) E yt(p,t). The urgency U of a transition is depicted as a number near the transition. The time constraints yt of an arc are depicted as an interval on the arc.
The operational semantics of TaPN-nets is defined by incorporating time constraints into the firing rules of PT-nets. Transition t is enabled in the marking m = <ms,mt,mu>, if t is enabled in <S(TaPN),ms> and time constraints of t are satisfied, i.e., each token a from a place p involved in the firing of tsatisfies mt(z) E yt(p,t).
An execution of a TaPN-net is a sequence of steps of the following two kinds. A transition firing step is the firing of enabled transition t that consumes involved tokens from places •t and produces new tokens to places t»; the urgency clock of t is set to zero, mu(t) = 0, and, for each produced token a, the clock value is set to zero, i.e., m(a) = 0.
A time elapsing step corresponds to the elapsing S time units in each clock of the marking m. We assume that all token clocks and transition clocks run at the same pace. We denote by m+S the
marking with all its clocks increased by 8, i.e., for each token a 6 ms : (mt + 8)(a) = mt(a) + 8, and for each transition t : (mu + 8)(t) = mu(t) + 8. Under urgency restrictions, a time elapsing step 8 is allowed if there are no 8' 6 [0,8) such that the m + 8' marking has urgent transitions.
3. Overapproximation of the number of active timers
Coverability analysis or approximation using continuous Petri nets are classical examples of overapproximation in Petri net analysis. TaPN-nets with urgency are Turing-complete, and DP-systems on metric graphs are, obviously, less expressive (consider DP-systems dynamics monotonicity) [23]. Thus, we consider a restricted class of TaPN-nets, such that token-age constraints on input arcs are positive intervals of zero length and each transition t has zero urgency, i.e., U(t) = 0.
In metric graphs, if points are met in a vertex simultaneously, they are coalesced, i.e., the resulting effect is the same as if only one point came to the vertex. This peculiarity of metric graphs functioning hinders direct modelling of TaPN-nets using metric graphs. However, under the provided restrictions on TaPN nets, it is possible to overapproximate the number of different timers present simultaneously in a TaPN-net.
Let TaPN be a marked TaPN-net and m = <ms,mt,mu> be its marking. Let us introduce an equivalence relation ~x over tokens:
Vai, a2 6Tok(m): ai ~T a2 ^ ai 6 ms(p)A a26 ms(p)A mt(ai) = m(a2) where tokens ai and a2 are ~t -equivalent iff both are located in the same place p and have the same clock values.
When the clock value of a token a located in place p passes the maximum value of all the time constraints on outgoing arcs of p, it cannot further involve in firings and its clock becomes redundant. Such tokens are dead [16] by time restrictions and we may remove their clocks from consideration. We denote by ActiveTokens(m) the set of active tokens in marking m
ActiveTokens(m) = {a 6 Tok(m) | a 6 ms(p) Л mt(a) < maxv<p,t>eF(yt<p, t>)} The quotient set Timers(m) = ActiveTokens(m)/~T consists of the classes of live tokens in marking m; each equivalence class corresponds to a set of tokens located in a place p that have the same clock value. In a physical system, a set of agents represented by such a class can share the same timer/external signal. The cardinality of \Timers(m)\ is the number of timers simultaneously active in the system.
Let TaPN = <<P, T, F, y>, yt, U> be a TaPN-net, where P = {pi...p„} and T = {ti...tm}, and let mo be its initial marking. We construct DP-system Pr(TaPN), which will be used to overapproximates TaPN, with simultaneously constructing relation pb^Timers(m) xPr between timers Timers(m) and points in Pr(TaPN) created to approximate these timers.
We start by adding a vertex vi to Pr(TaPN) for each transition tt6 T. In Figure 3, an example of TaPN-net TaPN and Pr(TaPN) that overapproximates TaPN is given. Then, for each pair of transitions ti and tj in T and place pk in P such that there are edges <t, p> and <pk, tj> in F, we add an arc <v,-, v/> to Pr(TaPN) with length l(<v,, vj>)=y(<pk, tj>). For each place pk of P, for each token a in mo located in pk, for each two transitions ti and tj, such that arcs <t, p> and <pk, tj> are in F, we put a new point p at the beginning of arc <v, v/> in Pr(TaPN) and add <[a]~r, p> to фь. Technically, we don't put p to vi but on <vi, v/> as puttingp to vt would produce tokens to all outgoing arcs. For each marked place pk in P that has only outgoing arcs, for each transition tj in T adjacent to pk, we introduce a new vertice vo and a new arc <vo, v/> into Pr(TaPN) with length l(<vo, v/>) = f(<p, tj>); we put a point p to vo (as for pi and ti in Fig. 3) and add <tm, p> to фь, where tm = <p,, o>. The intention is that each potential firing of a transition tj in TaPN, producing tokens to its output places, corresponds to an event when a point in Pr(TaPN) reaches vj, generating points on its outgoing arcs. In Figure 3, two active tokens oscillate between places p2 and ps in TaPN, while their corresponding points in Pr(TaPN) oscillate on edges between v2 and vs.
Fig. 2. TaPN-net TaPN and corresponding DP-system Pr(TaPN) We say that Pr overapproximates TaPN-net TaPN, if for any TaPN execution a = mo^slmi^s2 m2..., for any state mt £ a and global time t £ R+\Co// that corresponds to m, \Timers(mi)\ < Npr(t) holds.
Lemma 1. Let TaPN be a TaPN-net with intervals of zero length as constraints on input arcs and each transition t has zero urgency. Let Pr(TaPN) be a DP system constructed from TaPN. Then Pr(TaPN) overapproximates TaPN-net TaPN.
Let a = mo^s1mi^s2m2... be an execution of TaPN. For each mtand corresponding time t, we define relation c Timers(m) x Pr. For initial marking mo and time point t = 0, yo is equal to yb. Let yt is defined for m-i and t-i is global time corresponding to m-i. Let si be a time elapsing step for 8 time units, and Ti=T-i+8 is the time corresponding to mi.
For each place pk, for each token a £ m-i located in pk, for each two transitions ti and tj such that there are edges <ti, p> and <pk, tj>, if (m-i + 8)t(a) < yt(<pk, tj>), add <tm, p> to yzi, where point p is located on edge <vi, Vj> at coordinate x(p) = (m-i + 8)(a) and tm = <pk,(m-i + 8)(a)>. If (m-i + 8)t(a) > yt(<pk, tj>), we don't add anything to as it means that the age of a won't let it be involved in a firing of tj.
Let si be a transition firing step for transition tj and Ti = t-i as time doesn't flow during a transition firing step. All pairs related to all tokens consumed by the firing are not more in For each new token a £mt in place pk produced by the tj-firing for each two transitions ti and tq such that there are edges <ti, pk> and <pk, tq>, we add <tm, p> to yzi, where point p located at the beginning of edge
<vi, Vq>, and tm = <pk, 0>.
Note that for each marking m, for each active token a, image yf[a]^) is not empty as, when all the pairs related to a cease from a is dead and [a]~T ceases from Timers(mi) by definition. For different equivalence classes tmi = <pi, Ti> and tm2 = <p2, T2>, images yftmi) and yfm) are disjoint as they either lie on different graph edges if pr£p2, or lie on different coordinates if Ti f T2. Hence, \Timers(m)\ <npt(t).
Now let us consider a class of TaPN-nets without urgency and with inscriptions [x, &>) on input arcs of transitions. Now a token a may be involved in a firing of transition t, even if its clock passed maximal interval lower bounds at time t and t is disabled. When some other token ao comes to an input place p of t and ao clock satisfies inscription on arc <p, t>, transition t may become enabled and may fire consuming a. Such a token a is not dead at t but we call it passive. For such TaPN-nets, set of active tokens ActiveTokens(m) is defined as follows
ActiveTokens(m) = {a £Tok(m) \ a£ms(p) A m(a) < max {x \ <p,t>£F & yt<p,t>=[x,x)} } Theorem 2. Let TaPN be a TaPN-net without urgency, with inscriptions [x, <x) on input arcs of transitions, and with initiai marking mo. Let Pr(TaPN) be a DP-system constructed from TaPN. Then Pr(TaPN) overapproximates TaPN-net TaPN.
Let t be a transition and its firing produces tokens a ¡...an to output places of t. As inscriptions are unbounded, t may fire not necessary immediately at its earliest enabling time t but later, at time to = t + 8. Due to the monotonicity of Petri net firing rule and monotonicity of time constraints of form [x, <x), we may fire transition t at time t and then just keep (freeze) tokens ai... an at output positions of t until t + 8. Such execution may only increase the number of tokens in TaPN at interval [t, t + 8]. Henceforth, when we want to overapproximate execution a we may consider execution a' such that if a transition t will fire then it will fire at its earliest enabling time. Now we may apply the similar argument as in Lemma 1, and obtain | Timers(mi) | < Npr(T). As urgency may only narrow a set of possible executions of a TaPN-net, we get the following corollary.
Corollary 3. Let TaPN be a TaPN-net with urgency, with inscriptions [x, <x) on input arcs of transitions, and with initial marking m0. Let Pr(TaPN) be a DP-system constructedfrom TaPN. Then Pr(TaPN) overapproximates TaPN net TaPN.
'TaPN
Pi
Fig. 4. TaPN-net TaPN and corresponding DP-system Pr(TaPN) In Fig. 4, TaPN-net TaPN is on the left, and DP-system Pr(TaPN) that overapproximates TaPN is on the right. In [7], it was calculated that the growth of the number of points in Pr(TaPN) is
N(Pr(TaPN)) = t2 (ai + a2 + a3 + a4 + a5) / (2(ai + a4 + a5) (a2 + a3) (ai + a3 + a5)) + O(t) This gives us asymptotical estimate on the upper bound of the growth of number of active timers in TaPN.
In Pr(TaPN), the arrival of a point to a vertex always corresponds to emitting of new points on the outgoing edges. In TaPN, the firing of transition t in TaPN may occur only when all input places •t have tokens with suitable clock values. Thus, some arrivals of points in r do not represent real firings of transition t in TaPN. In addition, tokens do not collapse in TaPN-nets; thus, we overapproximate the number of different clock values in TaPN. However, the number of different clock values is an important parameter for a system when, for example, its components use shared clocks.
4. Simulation of DP-systems using timed-arc Petri nets
To simulate DP-systems using Petri nets with real-time semantics, we need to represent points moving along edges of a graph using clocks in a Petri net. The advance of a point along an edge in a metric graph is modelled with the progress of the corresponding clock in a Petri net. The number of points on a metric graph may grow indefinitely when edge lengths are incommensurable; therefore, it is not possible to model the evolution of a system of points on a metric graph using clocks in classical time or timed Petri nets [2, 24] because these models have finite structurally-determined number of clocks. On the contrary, in timed-arc Petri nets, clocks are
assigned to tokens [25], and the number of clocks can grow along with the number of tokens in the net; therefore, TaPN-nets with token-based time semantics suits well for simulating DP-systems.
Fig. 5. A metric graph r and a fragment of TaPN(r) built from edge ei Let Pr be a DP-system and TaPN(Pr) be the resultant TaPN-net simulating Pr. For each edge e of r connecting vertices a and b, we add two places eab and eba, where eab models arc <a, b> and eba models arc <b, a>. In Fig. 5, DP-system Pr is on the left, and a fragment of its simulation TaPN(PF) built for edge ei of F is on the right. Each point p on edge <a, b> of Pr such that x(p) = xo is represented with a token a inpab such that m(a) = x0. In Fig. 5, two points on edge <vi, V5> at the distances 3 and 5 from vi are modelled with two tokens that have timer values of 3 and 5, correspondingly. In addition, we add transitions tab and tba with U(tab) = U(tba) = 0, which firings model events when a point on e reaches b and a, respectively. We add arc <eab, tab> with yt(<e„b, tab>) = l(e), and, for each edge et in the metric graph connecting vertex b with vertex c, we add arc <tab, ebc> with yt(<tab, ebc>) = 0. Also, for each place eab, we add transition a with U(tab) = 0 (as for e5i in Fig. 5); tab models collapsing of points in a vertex by consuming two tokens with clocks equals to 0 in eab, and puts only one token with a clock equal to 0 back.
The tool support and details of the translation software implementation are provided in [26]. The translation enables us to utilise well-known analysis tools for TaPN nets - TAPAAL/UPPAAL[29] and ReNew [30], to conduct behaviour analysis and numerical experiments for metric graphs. In Fig. 6, the translation of a metric graph DP to TaPN-net in TAPAAL representation using the developed tool is demonstrated.
Fig.6. Tool support for translation of metric graph to TaPN-net For example, the following properties can be checked using TCTL engine of TAPAAL:
- is it possible that more than 50 signals come to node X in 10 seconds;
- is it possible that in the next 5 minutes two messages come with the difference of their time moments less that e milliseconds;
- does it hold, for the system initial phase with duration of 15 minutes, that if a point comes to node X, then no points comes to node Y within 10 seconds,
etc.
5. Conclusion
In this paper, we studied correspondence between metric graphs and a restricted class of TaPN-nets. We demonstrated that, for a restricted subclass of TaPN nets, it is possible to overapproximate the number of active timers using DP system on a metric graph. Such a correspondence connects studies on TaPN-nets and metric graphs, which allows to conduct analysis of the number of different timers in a given subclass of TaPN-nets using a system of points dynamics.
Our current research is to further weaken the suggested restrictions on TaPN-nets and, simultaneously, to find asymptotics for a more general class of DP-systems. Current working hypothesis is that the approach used to obtain estimates in [15, 16] could be extended to arbitrary directed graphs letting the suggested overapproximation to be used for a wider class of TaPN-nets. Conducted computed numerical experiments support it.
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Information about the author / Информация об авторе
Leonid Wladimirovich DWORZANSKI - doctor of natural sciences (Doctor rerum naturalium), independent researcher. Research interests: analysis methods for discrete event dynamical systems, models for parallel and distributed computation, models for real-time systems, well-structured transition systems.
Леонид Владимирович ДВОРЯНСКИИ - доктор естественных наук (Doctor rerum naturalium), независимый исследователь. Сфера научных интересов: методы анализа динамических свойств дискретных динамических систем, моделей параллельных и распределенных систем, моделей систем реального времени, вполне структурированных систем переходов.
