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Published in IET SoftwareReceived on 9th January 2009Revised on 27th May 2009doi: 10.1049/iet-sen.2009.0002

In Special Issue on Performance Engineering

ISSN 1751-8806

Transient calculations on process algebraderived Markov chainsA. Clark S. GilmoreLFCS, University of Edinburgh, Edinburgh, UKE-mail: [email protected]

Abstract: The process of obtaining transient measures from a Markov chain as implemented in the software,ipclib is described. The software accepts models written in PEPA, Bio-PEPA or as a Petri net. In the case of theprocess algebras, a rich query specification language particularly well suited for the derivation of passage-timequantiles is provided. Such measurements are obtained from the derived Markov chain through a processknown as uniformisation. The authors detail how the process algebra and query specification language allowone to ensure that the passage-time calculation is valid and then the entire process through to the finalcalculation of the cumulative distribution and probability density functions of the passage in question. Theauthors also show a more generic transient measure for which the full probability distributions at specifictimes are required.

1 IntroductionWhen analysing a performance model of a system a commonquery is over the duration of a given passage within themodel. In a client–server style model, this is often theresponse time, that is the time that a client must wait fortheir request to be satisfied by the server. In other modelswe are simply analysing the time taken between two eventsor circumstances. Where the model in question is acontinuous time Markov chain (CTMC) or a representationfrom which a CTMC may be derived, the average passageduration may be computed from the steady-state probabilitydistribution obtained by solving the CTMC.

In general CTMCs are well suited to the analysis ofaverage behaviour, derivable from the steady-stateprobability distribution. This paper is concerned withtransient analysis of models that compute measures that arenot time independent. Essentially we wish to analyse theprobability distribution of a model a given time after someevent or from the initial configuration. Here we describesoftware support for the computation of passage-timequantiles computed via a technique known asuniformisation. A passage-time quantile is simply a pointtaken along the cumulative distribution function (cdf) ofthe passage in question. The cdf maps time (usually along

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the x-axis) against the probability of completing thepassage at or before that time. This allows the modeller toanswer such questions as: ‘What is the probability that arequest is responded to within 4 seconds?’ It is also possibleto plot the probability density function (pdf) where the cdfis the integral of the pdf. The pdf maps time against theprobability density of completing the passage at exactly thattime.

This paper describes the calculation of passage-timequantiles/densities from a model which may be translated intoa CTMC. The technique used is known as uniformisation –though it is sometimes called randomisation. We show howuniformisation has been integrated into a generic software toolfor analysing models written in higher-level formalisms.Specifically we are chiefly interested in models written in thestochastic process algebra PEPA [1] and its derivativebiological modelling language Bio-PEPA [2].

1.1 Related work

Hydra [3] is another software tool used for extractingresponse-time profiles, in that case from Petri nets. Thefocus in Hydra is on analysing very large models, forresponse time this obviously depends on rate valuesinvolved (see Section 5.4) but it has been used to solved

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models with state-space sizes of the order of 106 and even107. One technique that is used is pre-compilation, themodel is input by the user and the Hydra softwaregenerates a program in standard C to generate the state-space and solve the generator matrix. This program is thencompiled and run. From the outset the ipc software hasbeen linked with the Hydra software; the ipc software cangenerate a Hydra model description from a PEPA model(and hence also from a Bio-PEPA model) or ipc can readin Hydra model description files. Recently some of thetechniques described in this paper have been implementedin the Hydra software, most notably the uniformiser nowproduces both the cdf and pdf of a passage at the sametime and the absorbing state is used to terminate theuniformisation computation (see Section 5.1).

The Mobius [4] project is aimed at delivering a multi-paradigm modelling software tool. As such there is somesimilarity with the ipc software, although we are mostlyconcerned with process algebras and in particular PEPAwhereas the Mobius software is not limited to processalgebras. In addition, Mobius, has several analysistechniques including discrete-event simulation. We havediscrete-event simulation and continuous simulationmethods available in the PEPA Eclipse Plug-in [5], butnot in the ipclib suite. The Mobius framework excels in thespecification of reward structures over the input model.These allow the definition of instant-of-time or interval-of-time rewards. Our framework here does not provide acomparable reward specification language. Additionallyworking with Sanders of the Mobius project, Dienercompleted a masters thesis [6] comparing severaluniformisation techniques.

The MRMC (Markov Reward Model Checker) [7] projecthas produced several results for transient analyses of Markovchains. Here the problem is stated as time-boundedreachability – and amounts to determining the probabilityof reaching a set of target states within a given time bound.In [8] the authors report on the efficient detection ofsteady state during transient analysis. When performing atransient analysis over a large time span – suppose the userhas asked for the time series analysis for a given timebound – it may be that the model reaches steady state oran equilibrium before the end of the time bound. Whenthis happens the software can stop calculating for furthertime points since they will all have the same probabilitydistribution. However, for a passage-time analysis this isinappropriate since we modify the Markov chain to have anabsorbing state such that we measure only the first passageand not subsequent passages. Although we focus onpassage analysis our software can also be used for suchtransient analysis (see Section 6) and there suchoptimisation is certainly appropriate.

The Prism [9] model checker implements a uniformisationalgorithm in order to model check transient properties ofCTMCs. The software can be used to check properties

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written in Continuous Stochastic Logic (CSL) [10]. Ouripc software is capable of translating process algebra modelsinto an input suitable for the Prism model checker, thusallowing the user familiar with CSL the opportunity tospecify their queries in their known setting. By using ipc togenerate the Prism model the users benefit from theadvantages of compositional modelling and in particularwith respect to the constraints imposed on passagesdiscussed here in Sections 3 and 4.

Structure: The paper is organised as follows in Section 2 wefirst give a brief introduction to PEPA, the process algebrain which our example models are written. We then givetwo example models which we will use to illustrate thetechniques implemented in our software. In Section 3 wedetail the constraints imposed on the passage-time query inquestion by the use of the uniformisation algorithm. Theconstraints refer to the transitions within the Markov chainin relation to the specified passage. Then in Section 4 weexplain how the passage is specified by the user where theuser is concerned with the model at the level of the processalgebra but the uniformisation routine expects a Markovchain and source and target sets of states. We demonstratehow the use of a compositional process algebra allows us toensure that the query the user specifies is always translatedinto a valid passage-time query which does not violate anyof the imposed constraints. In addition we discuss someoptimisations that are made possible through the use of aprocess algebra to derive the underlying Markov chain.Section 5 provides a full description of the uniformisationtechnique in order to obtain passage-time quantiles from aMarkov chain for a passage specified as a set of source anda set of target states and provides results for our firstexample. Our second example is then analysed in Section6, which is concerned with the calculation of more generaltransient measures in which probability distributions atspecified times are calculated. In particular, this allowsthe production of a time series analysis. We finish bydetailing our implementation in Section 7 and concludingin Section 8.

2 Example modelsWe illustrate the tools and techniques in this paper byreferring to two examples. We draw one example from thearea of computer service analysis and the other fromsystems biology. We must first briefly introduce ourfavoured process algebra, PEPA.

2.1 PEPA

We use the popular process algebra PEPA to compositionallydescribe our models. A model described by a PEPAdescription has an underlying Markov chain representation,although the user is hidden from the details of theunderlying states. Each defined sequential component is adescription of a small stateful process and each such iscombined using the cooperation combinator with the

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restriction that the two must synchronise over the specifiedaction labels. This may mean that some states areunreachable so composition does not always increase thestate space but in general the state-space size does increaserapidly. The PEPA language has the following combinators

The process term (a, l):P describes a process that mayperform the action a at rate l to become the process P.The rate may be a numerical rate or the special rate `,which means that the operation is performed passively bythis process that must be subsequently synchronised withover this activity. The other process involved in thesynchronisation determines the rate of the activity. Theprocess P1 þ P2 depicts competitive choice between theprocesses P1 and P2. The operator is cooperation/synchronisation between two components over the given setof actions L. The special synchronisation specifies thatthe two components synchronise over all activities that bothmay perform.

A model is represented by a series of definitions thatdescribe the sequential behaviour of named components.These named components are then combined together in amain system equation, which represents the interactionbetween the various components in a model. Full details ofthe PEPA stochastic process algebra can be found in [1].

2.2 Layer-7 two-way architecture

Fig. 1 depicts a layer-7 two-way service architecture. A layer-7 two-way architecture allows the service to provide acommon front to several different kinds of request. Eachrequest is sent to an administrator who analyses the requestand then dispatches it to be serviced by the appropriatebackground server. Each background server awaits suchspecific requests and returns them to the administrator whoforwards the response to the appropriate client whoinvoked the original request. Fig. 2 shows the PEPAmodel description of a particular layer-7 two-wayarchitecture configuration. By analysing such a system wecan analyse the response times of different kinds of requests

Figure 1 Structure of a layer-7 two-way architecture

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as well as in general. We could also modify theconfiguration of the service to enable us to decide, whichwill perform better under different conditions. For example,we can vary the rates of requests or the capacity of eachkind of server, which in reality would result from server re-deployments or upgrades. We will see the passage-timeanalysis of this model in Section 5.2. In the next sectionwe discuss how this process algebra model allows us toderive a Markov chain in a form suitable for the calculationof passage-time quantiles measuring the time from theclient’s request to their received response.

2.3 Michaelis–Menten

We show here the PEPA model of a chemical reactionbetween an enzyme E and a substrate S to produce anintermediate compound E:S and finally a product P. Thereactions are modelled with Michaelis–Menten semantics.

The four chemical species react over three reactionchannels: r1 converting E and S to E:S, the backwardreaction r�1 taking E:S to E and S and the reaction r2

converting the compound E:S into product P and releasingthe enzyme E. The reaction rates are governed by kineticlaws involving rate constants (k1, k�1 and k2) and themolecular counts of the species involved.

The reaction is initiated with a quantity of enzyme and aquantity of substrate and over the course of time thereactions will convert the substrate into the product(although many forward steps to the intermediatecompound via reaction r1 may be rapidly undone by thereverse reaction r�1). However, reaction r2 is not reversiblein this model and so once product is produced by thisreaction it cannot be destroyed by any other reaction.

Crucially, the rates of the reactions are functions of themodel state, not constants. Thus in the initial state onlyreaction r1 can fire (to form the compound). When somecompound (E:S) is present then reactions r�1 and r2 canfire, otherwise not. This switching behaviour is controlled

Figure 2 PEPA model of a layer-7 two-way architecture

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by the kinetic laws, which use the number of molecules ofeach species as a multiplier (giving a zero rate when thespecies is not present, and thereby switching off certainreactions). This model reaches a deadlock state in whichthere is no free substrate and no free compound. In thisstate reaction r1 cannot fire (because it requires thepresence of the substrate) and reactions r�1 and r2 cannotfire (because they require the presence of the compound).

The Bio-PEPA species definitions indicate whichchemical species participate in each reaction and whetherthey are reactants consumed by the reaction or productsproduced by the reaction. The symbol # is used for theformer (because the molecular count of that chemicalspecies goes down) whereas the symbol " is used for thelatter (because the molecule count of that chemical speciesgoes up). The model equation of a Bio-PEPA modelspecifies which species participate in which reactions andwhether they are initially present (denoted by a 1 in themodel equation) or initially absent (denoted by a 0 in themodel equation). The Bio-PEPA model of this system isshown in Fig. 3.

The PEPA model shown in Fig. 4 is derived from theBio-PEPA input. It is presented in reagent-centric style[11], meaning that it indicates the direction of the reactionas a change from higher to lower quantities of thecompound. If the reaction increases the quantity of thechemical species X (" in Bio-PEPA) then the derivedPEPA model will have a change from X low to Xhigh,meaning that the molecular count of X is higher after thereaction. If the reaction decreases the quantity of thechemical species X (# in Bio-PEPA) then the derivedPEPA model will have a change from Xhigh to X low,meaning that the molecular count of X is lower after thereaction. With only the direction of change known (fromhigh to low, or vice-versa), it is possible to derive exactlythe system of ordinary differential equations that specifiesthe continuous limit of this reaction behaviour [12]. Oursoftware converts the derived PEPA model into a CTMC,which can then be solved for the steady-state distributionand related measures. However, in this case this isinappropriate since eventually all of the substrate isconsumed and only enzyme and product remain, at thispoint the model is in a deadlocked state. We instead use

Figure 3 Bio-PEPA model for Michaelis–Menten reactions


transient analysis of the computed Markov chain to enablea time-series plot allowing us to compute the time takenfor all (or a portion of) the substrate to be consumed. Weperform this in Section 6.

3 Passage time constraintsIn this section we detail the constraints imposed on the formof the passage within the Markov chain by the uniformisationmethod of obtaining passage-time quantiles. In order toperform uniformisation on a (derived) CTMC we mustspecify two sets of states: the set of source states S and theset of target states T. The passage set is the set of all statesthat lie on some path between the source and target setsincluding the source set but not including the target set.The intuition is that a source state is reached by an eventwhich begins the passage, hence the sojourn time of asource state is included in the time taken to complete thepassage. A target state is reached by an event which endsthe passage, therefore the passage is completed when atarget state is entered and its sojourn time should not beincluded in the time to complete the passage.

In order for our uniformisation calculation to be valid, itmust be the case that no state within the passage set maytransition to a state within the source set. The reason thatthis is invalid is that we must obtain the probabilitydistribution of the source states at the beginning of thepassage. If there is no transition from the passage set toone of the source states then this can be computed usingthe steady-state probability distribution of the embeddedMarkov chain to calculate the frequency with which eachsource state is entered relative to the frequency that thesource set in general is entered. Where such a transitiondoes exist then this calculation will not provide theprobability distribution we require since it is possible toenter particular source states without beginning the passageof interest.

Although this would appear to restrict the set of models forwhich passage times may be calculated any model thatviolates this may be turned into an equivalent model that

Figure 4 Translated PEPA model for Michaelis – Mentenreactions


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does not by duplicating the source states that may be returnedto from within the passage set. The cost of thistransformation is to increase the state-space size; however,the increase is linear. In fact, we must increase the statespace by at most the size of the source set.

Note also that we cannot transition from a state within thepassage set to a state outside both the passage set and thetarget set. Such a state either has a path back to the passageset and hence to the target set and is hence within thepassage set (and as such may not transition into the sourceset). Alternatively the state is (or has a path to) adeadlocked state. Such deadlocked states are not allowed ifthere are multiple source states, since then the steady-stateof the embedded Markov chain cannot be computed. Ifthere is only one source state this is not required, however,we view such a deadlocked state as within the passage set.This also means that there is a non-zero probability thatthe passage of interest is never completed at all. Morestrictly then the passage set is defined as all those statesthat are reachable from the source set without passingthrough the target set.

4 Compositional modelsThe technique of uniformisation is defined over theunderlying CTMC and not in terms of the higher-levelformalism used to define the CTMC. However, forpassage-time quantiles we must specify to theuniformisation routines the source and target states and thismust be done in the higher-level formalism since the usershould not be required to understand the derived statespace of the model. In this section we concentrate on theprocess algebra PEPA and how our software with its queryspecification language allow a valid passage-time analysis tobe presented to the uniformisation routine without userintervention.

When designing a query specification language which willbe used for passage-time queries, one must consider twofactors. Firstly can the compositional nature of both themodel and the query be used to ensure that the constraintsas specified in Section 3 are not violated? Secondly can thecompositional nature be used to optimise the calculation ofthe results?

Two techniques are commonly used to describe the statesalong the passage of interest. Firstly state-based techniquesrequire the user to specify the states of interest using thelocal states of the separate processes or tokens as a filter onthe entire state space. Alternatively action or eventsequences that result in the source or target states aredescribed. Measurement specification is still an active areaof research and a recent paper by the current authorsdescribes a specification language which blends the use ofactivity observations with state-based filters [13].

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4.1 Stochastic probes

To describe passages we insert probe components thatobserve activities performed by the rest of the model, viapassive synchronisation, and change their local stateaccordingly. We then use the states of the probecomponents as a filter on the full state space of the model.Probe components can be manually composed and addedto the model or specified in our (regular-expression-like)language, which is then automatically converted into PEPAcomponents and inserted into the model.

The major rule that we must respect is that no state alongthe passage may have a transition outside of the passage or atransition which targets a source state.

For the purposes of this paper we will assume that themodel performs a ‘start’ activity when and only when themodel enters a source state. Similarly the model performs a‘stop’ activity when and only when the model enters atarget state. Elsewhere [14] we describe how such start andstop signals can be generated automatically when stochasticprobes are added to the model in such a way that thebehaviour of the model is not affected. We can thereforeadd a simple passage probe, which is running whenever themodel is within the passage set.

Because the passage-probe may only be stopped by themodel entering a state within the target set, it is notpossible to transition out of the passage set. In the casethat this was possible for the original model, such a state(outside of the passage set) is duplicated as the same statedescription but with the passage-probe component eitherrunning or not. This ensures that the first property ismaintained. The passage probe can be defined andcomposed with the main system by

PassageStopped ¼ (start, `):PassageRunningþ (stop, `):PassageStopped

PassageStopped ¼ (start, `):PassageRunningþ (stop, `):PassageStopped

PassageStopped Systemwhere L ¼ {start, stop}

We can, in a similar fashion, add a second probe called asource probe, which is in the running state only whenthe model is in a state directly following the start of thepassage. Any source states that may be revisited during thepassage are duplicated in a similar manner to the above,and hence the duplicated source state is not considered tobe within the source set for the purposes of the passage-time calculations. The definition of the source probedepends upon the alphabet of activities performed by theentire model; for one in which the alphabet is just{start, stop, a, b} the source-probe is defined and composed

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with the main system by

SourceStopped ¼ (start, `):SourceRunningþ (stop, `):SourceStoppedþ (a, `):SourceStoppedþ (b, `):SourceStopped

SourceRunning ¼ (start, `):SourceStoppedþ (stop, `):SourceStoppedþ (a, `):SourceStoppedþ (b, `):SourceStopped

SourceStopped Systemwhere L ¼ {start, stop, a, b}

Astute readers should note that the first line of theSourceRunning definition is not required since the ‘start’activity should not be performed once already in a sourcestate but it is included for completeness. By adding simpleprobe components, which may be specified with an evensimpler probe language, we can ensure that any necessarystate duplications are made automatically during state-spacegeneration. We also do not duplicate states where this isnot necessary to stay within the constraints of themeasurement.

Finally one might ask: What happens if the initial state isalong the passage? In this case some states along the passageare duplicated during state-space generation, which mayimpact on how long it takes to solve the model. However,the results will not be affected. More specifically the statesthat lie on a path from the initial state to the target set willbe duplicated and become transient states (states which arenot visited in the long run). Since a passage-timemeasurement is not dependent on the initial state of thesystem, the user can always re-write the system equation toavoid this, but we stress that it will make no difference tothe results obtained only how quickly they may be obtained.

4.2 Optimisation

As will be described in Section 5 in order to make sure thatonly the probability of completing the first passage and notsubsequent passages are computed, the Markov chain ismodified so that all target states have only one outwardtransition whose destination is an added absorbing(deadlocked) state. This means that there are some states inthe modified Markov chain which are unreachable from thesource set. These are the states that are not contained inthe passage set or the target set, we will call this set U.

In addition, we will never query the probability mass of anystate within the unreachable set and the states in this setcannot affect the probability of other states and hence thepassage-time quantiles we are computing. We must stillderive this set as it will be used to calculate the seeding ofthe source states, that is the probability distribution at thebeginning of the passage when all the probability mass iswithin the source set. However, once this is done these


states may be removed for the rest of the computation.This would be done at the same time that the Markovchain is modified to have an absorbing state to which alltarget states must transition. As we will see the rest of thecomputation involves (possibly many) matrixmultiplications in which the modified Markov chain ismultiplied by a probability distribution. Therfore if themodified Markov chain can be made smaller by removingthe states in the unreachable set U, then it is a potentiallylarge saving in both the space and time required for thecomputation.

Using the process algebra and probe setting describedabove, this set is easy to calculate; it is the set of all statesin which the passage probe is in the ‘stopped’ state minusthe target set. The target set itself may be computed usinga probe that is similar to the source probe. The targetprobe can be defined and added to the main system by

TargetStopped ¼ (stop, `):TargetRunningþ (start, `):TargetStoppedþ (a, `):TargetStoppedþ (b, `):TargetStopped

TargetRunning ¼ (stop, `):TargetStoppedþ (start, `):TargetStoppedþ (a, `):TargetStoppedþ (b, `):TargetStopped

TargetStopped Systemwhere L ¼ {start, stop, a, b}

Note that it is the same as the source probe with the roles ofthe ‘start’ and ‘stop’ activities reversed. This method, though,has the disadvantage that the target states may beunnecessarily duplicated since there was initially norestriction on a state looping back to a target state. That is,there is no reason why a state outside the passage set(including one within the target set itself) may nottransition into a state within the target set, but this probewill distinguish such occurrences of the target states. Thiswill only impact the calculation of the seeding of the sourcestates since the duplicated target set states will be withinthe unreachable set. This is therefore entirely harmlessif the source set has only one member since the seeding ofthe source states can be avoided when there is only onesource state (it is the trivial seeding in which all of theprobability mass is held by the single source state).

Regardless of how the target set and the unreachable setare identified, with a compositional approach to buildingthe model it is straight-forward to see what kind of savingwe can make by removing the unreachable set. This ofcourse depends on the measurement but commonly we aremeasuring the response time as observed by a singlecomponent. Suppose there are two clients and one server.Each client may ‘request’ something from the server whichwill subsequently ‘respond’ to the client. The server willpassively wait for all incoming requests and make a


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response after an exponentially distributed delay. In this caseour measurement is begun with the first client, making arequest and stopped when it receives its response. Since thestate of the second client is independent of the state of thefirst client, there are as many states outside the passage-setand target-set as there are within. If we increase thenumber of clients this still means that the state space issplit approximately equally by those states within thepassage and target sets. So the unreachable set is roughlyhalf the size of the model. It gets larger if we add states inwhich the clients can sojourn when they are not waiting onthe server. In fact, the state space of the whole model canbe divided into N equally sized sets where each setcorresponds to a particular local state of the first client. Anumber of these sets, say t , N , are within the passageand target sets and some do not depend on the state of thefirst client. So in this example the unreachable set is(N � t)=N times the size of the entire state space. Thisclearly may be well over half the size of the state space.Such a simple model can be complicated by competitionfor a scarce resource such as a server which only accepts alimited number of concurrent requests. This complicatesthe calculation somewhat, but in general we find that halfis a good estimation as to how much of the state space is inthe unreachable set and can therefore be removed from themodified Markov chain before any matrix multiplicationstake place.

5 UniformisationThis section details the steps used to derive the cdf (and/orpdf) of a passage within a model represented as a CTMCin which we have specified a source and target-set and arecomforming to the constraints laid out in Section 3. Wefirst detail the prerequisites:

1. The CTMC may be represented by the generator matrixQ. The generator matrix is an n� n matrix where n is thenumber of states in the Markov chain. Each rowcorresponds to one state and the value in one cell of a rowcorresponds to the rate at which the Markov chain maytransition from the state given by the row number to thestate given by the column number. The diagonal values aregiven by subtracting from zero the sum of the other valuesin the row. Hence if we write r (i, j) to mean the rate atwhich the Markov chain may transition from state i to jthen the generator matrix Q is written as:

Qi, j ¼r(i, j), i = j0� (

Pnk¼1 r(i, k)), otherwise

�

This requires that for any state i, the rate r (i, i) is zero – thiscondition is usually stated by insisting that the modelcontains no self-loops.

2. The generator matrix may be solved to obtain the steady-state probability distribution p where pi is the long-term


probability of being in state i. This requires that the modelbe deadlock free.

3. The set of source states S and the set of target statesT. The probability at time t that we compute is theprobability of moving from any of the states in S to any ofthe states in T within time t.

The steps in the computation of the pdf and the cdf of aparticular passage within a CTMC are summarised asfollows:

1. Uniformise the generator matrix to obtain the new matrixP by

P ¼ Q=q þ I

where Q is the generator matrix, I is the identity matrix and qis a rate value which is chosen to be greater than themagnitude of all of the rates within the generator matrixincluding the values along the diagonal. Therefore we haveq . maxij jQij j which can be reduced to q . maxi jQiij

since the magnitude of the diagonal values in each row arethe sums of the other values in the row which cannotbe negative. Since q is of greater magnitude than any of the(negative) diagonal values dividing by q returns a negativenumber x : �1 , x , 0. This means that adding theidentity matrix ensures that all rate values are positive.

The bottom graphs of Fig. 5 depict the probabilities ofperforming a n hops within time t for a fixed rate q. Notethat these graphs do not depend on the passage beinganalysed only on the value of q.

2. Add to this uniformised matrix P an absorbing state. Thisstate has no out-going edges to any state other than itself,which it loops to with probability 1.

3. Modify all target states (states in T ) to transition withprobability one to the absorbing state. Call this new matrixP0. The reason for our absorbing state is to ensure that wecompute the probability of the first passage and notsubsequent completions of the passage. That is, if we are instate i [ T at time t then we know we are completing thepassage at time t and it is not the case that we completedthe passage at some earlier time and remained in orreturned to state i.

4. Compute the probability distribution at the start of thepassage, p(0). That is the probability, in the originalMarkov chain, of being in each particular source state giventhat we must be in one of the source states. This is doneusing the embedded Markov chain from the originalMarkov chain Q. We obtain p(0) by

p(0)k ¼

0, k � Spk=pS , k [ S

�

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Figure 5 Top graph shows the probability of completing after n hops, graph on the bottom left shows the probability ofcompleting n hops within a given time t, whereas the graph on the bottom right shows the probability of performing agiven number hops against t

2

where pk is the steady-state probability in the embeddedMarkov chain of being in state k and pS is the steady-stateprobability in the embedded Markov chain of being in anyone of the source states, that is

Pk[S pk. Where there is

exactly one source state then the steady-state probabilitydistribution need not be calculated and p(0) is given by

p(0)k ¼

0, k � S1, k [ S

�

since for the one source state j, pj ¼ pS .

5. Compute the probability distribution after n hops of theuniformised Markov chain; given by p(n) where p(nþ1)

¼

p(n)P 0. The top graph in Fig. 5 is an example passagecalculation showing the probabilities of being in certainstates after a given number of hops. In particular, theprobability of being in a target state is the probability thatthe passage is completed in exactly n hops while theprobability of being in the absorbing state is the probabilitythat the passage is complete in less than n hops. Theaddition of these two probabilities gives the probability ofcompleting within n hops.

6. To compute the cdf at each time t we finally adjust eachhop value by the probability that n hops are performed


within time t. Recall that each hop has the same averageduration of 1/q so the probability that we may perform nhops in time t is an erlang distribution. An erlangdistribution for a particular value of q is shown in thebottom two graphs of Fig. 5. So then our cdf at time t is:Pn¼1

n¼0 Er(n)t p(n)

Twhere Er(n)

t is the probability that the nthhop will be performed at or before time t and p

(n)T

is theprobability of being in any of the target states after exactlyn hops of the uniformised matrix, P0.

In the final step above we have computed the cdf of thepassage by multiplying the probability of being in a targetstate after exactly n hops by the probability of performing nwithin the time t. This probability is given by(1� e�qt Pn¼1

k¼0 (qt)k=k!). For the pdf we substitute this forthe probability of performing n hops at exactly time t. Thisis given by: qntn�1e�qt=(n� 1)!.

For completeness we provide the full formulae forcomputing the cdf and pdf of the passage, respectivelygiven by

F~ij(t) ¼

X1n¼1

1� e�qtXn�1

k¼0

(qt)k

k!

!Xk[~j

p(n)k

0@

1A


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and

f~ij(t) ¼X1n¼1

qntn�1e�qt

(n� 1)!

Xk[~j

p(n)k

0@

1A

A summary of the mathematical relationships involved in thisalgorithm is given in Table 1.

5.1 Producing passage-time quantiles

The above description of an algorithm for computingpassage-time quantiles is an ideal implementation with oneimportant flaw; the final step calls for a summation toinfinity. It is worth re-stating exactly the meaning of thissummation. By the final step we have the means tocompute an infinite number of ‘hops’ where each hop is aprobability distribution from which can be extracted theprobability of completing the passage in exactly n hops(p(n)T

). In addition, we know the probability of performingn hops in a given amount of time t since each hop has thesame average duration which is exponentially distributed,giving an erlang distribution for completing n hops withint time (Er(n)

t ). For any given n multiplying these two valuestogether gives the probability of completing the passage inthe given time t using exactly n hops. Since we do not carehow many hops it takes if we add together theseprobabilities for all values of n then we have the probabilitythat the passage is completed within time t.

Clearly, summing all of these probability values from zeroto infinity is impossible for a computer to do. However, therewill be some value X for which all values nX . X , theprobability of completing the passage within the given timein exactly nX hops is negligible. Hence at this point wemay stop computing probability values. Here we show twoconditions that suffice to find the value X.

Previously one method was to compute the probabilitiesfor successive values of n and whenever the probability((Pn((n))) was sufficiently low we assume that subsequent

Table 1 Relationships between the probability values

Probability of Value Name

completing thepassage in exactly nhops

Pk[S p

(n)k p(n)

T

performing n hopswithin time t

(1� e�qt Pn�1k¼0 (qt)k=k!) Er(n)

t

completing thepassage in n hops bytime t

p(n)T

Er(n)t Pn(t)

completing thepassage by time t

Pn¼1n¼0 Pn(t) cdf


values will also be sufficiently low. This method hasproblems when the passage we must complete has separatepaths that vary greatly in their length of hops. In thisinstance, it is possible for the probability to drop below thethreshold value but later climb above it. In this case thegiven method would stop calculating the probability valuesbefore they have a chance to rise above the threshold onceagain.

Our method is to monitor the probability of being in theabsorbing state after n hops – we designate this valueAbs(n). When this value climbs to within a suitablethreshold of 1 then there is no probability left to flowthrough the target states. Hence the probability of being ina target state for all values of n greater than the currentvalue must be below the threshold value, since thisprobability is multiplied by the probabilty of performing nhops within time t we know that all subsequentprobabilities will be below the threshold.

This method performs well; however, for small values of twe find that we compute more hop values than are required.This is because for small values of t it is unlikely that we areable to perform a large number of hops. However, if thepassage is long then it may be that the probability of beingin the absorbing state does not climb to within thethreshold of one until n is large – where by ‘large’ we mean‘larger than the number of hops we could hope to performwithin the time t ’. Therefore we also monitor the value ofPn(t) – the probability of performing n hops within timet – whenever this value falls below the threshold, we knowthat any subsequent values of n will yield negligibleprobability at time t [since Pn(t) is involved in the product]and hence we have determined a suitable value of X.

Our algorithm may be summed up by a recursive functionas

cdf (n, t)¼

0, Abs(n). (1� threshold)

0, Er(n)t , threshold

(p(n)T�Er(n)

t )þ(cdf (nþ 1, t))

, otherwise

8>>><>>>:

and similarly for the pdf function

pdf (n, t)¼

0, Abs(n). (1� threshold)


(p(n)T�Erp(n)

t )þ(pdf (nþ 1, t))

, otherwise

8>>><>>>:

where Erp(n)t is the probability of performing the nth hop at

exactly time t and is given by: qntn�1e�qt=(n� 1)!. Sincethere is a lot of shared computation our implementationcomputes both the cdf and the pdf of the passage together.

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5.2 Analysis results

The passage time results applied to our layer-7 examplemodel are shown in Fig. 6. In the model there were threekinds of requests, so we have analysed the response time ofeach kind of request separately as well as the response-timein general. These four analyses were specified with thestochastic probe definitions:

request1:start, response:stop



(request1jrequest2jrequest3):start, response:stop

These definitions were then translated automatically by oursoftware into PEPA components and attached to themodel. The software then derives the state space which isguaranteed to adhere to the constraints of the passage-timecalculation and the software already knows the source andtarget sets of states.

5.3 Technical points

We have shown how to compute passage-time quantiles fromcontinuous time Markov chains. However, we have left theactual numerical computation as given, although this isnon-trivial. For the cdf, we must compute

F~ij(t) ¼

X1n¼1

(1� e�qtXn�1

k¼0

(qt)k

k!)

!Xk[~j

p(n)k )

Notice in particular that we must compute k! for what may belarge values of k. In addition, we must compute qtk, also forpotentially large values of k. The large values here are in theorder of the number of hops, this value may be quite high – avalue in the order of thousands is not uncommon (in [15] thisnumber is said to be of the order of qt). Hence we can expectto encounter a problem with overflow. Even if some arbitraryprecision library is used (at a performance cost) computing

4The Institution of Engineering and Technology 2009

the cdf in this way is inefficient. Our first observation is that

(qt)k

k!is equal to:

Yi¼k

i¼1

qt

i

which allows us to avoid the computation of the large powerand factorial values.

Now for each value of n we must computePN

k¼0(qt)k

k!. We

need not compute each term separately. We can insteadcompute the infinite list of values by the recursive function

sumvalues(n, current) ¼ current : rest

where rest ¼ sumvalues nþ 1, currentþXN

k¼0

(qt)k

k!

!

Because our implementation is in the lazy programminglanguage Haskell we need not worry about the computationof an infinite list since we will only ever examine a finitenumber of elements from it. For a strict language thislaziness can be easily simulated. We now observe that eventhis computation does a large amount of re-computation.Namely the successive values of

PNk¼0

(qt)k

k!recompute all

previous values. However, we can use the same trick:

prodvalues(k, current) ¼ current : rest

where rest ¼ prodvalues kþ 1, current�qt

k

� �

This means that we can now update our sumvaluesfunction to take advantage of this. It now becomes a listtransformation function which takes in the list of productvalues computed by the above prodvalues function.

sumvalues(current, (n, p) : rest)¼ (n, current) : restsum

where restsum¼ sumvalues(currentþ p, rest)

We can also factor out the code to calculate the probability ofbeing in the absorbing state and/or a target state after exactlyn hops. Since otherwise we will recompute these values foreach time value we desire. Once we have factored out allthe common computation we have a set of infinite lists that

Figure 6 Results of analysing passages within the layer-7 model


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map n from n ¼ 0 to n ¼ 1 to values used in thecomputation of the cdf and pdf. We need only operate onthese lists for the values of t.

5.4 High rates

In this section we detail why the use of very high rates withinthe model can render the uniformisation technique unusable.Consider the very simple Markov chain shown in Table 2.

This model can be solved for the steady-state solution veryquickly because the number of states is very small. Using thesteady-state solution we can calculate the average time it takesto complete some passage, for example from the source statezero to the target state three. This gives an average passagetime of just over 20 time units. However, if we wish tocalculate passage-time quantiles for the same passage wemust uniformise the chain. Unfortunately, this involvesdividing through by the magnitude of the largest rate,which is slightly larger than 50 000. We already know thatthe passage takes on average 20 time units, so the numberof hops we will have to compute is at least: 50 000� 20,rather a lot for such a simple passage.

The same problem occurs whenever we have large ratesthat are involved in the passage, but the passage takesmuch longer than the reciprocal of the largest rates. Thismay occur because some small rates are involved in thepassage and/or because of looping within this model. Theauthors have encountered this problem in the modelling ofbiochemical reactions, in particular those with a Michaelis–Menten semantics which involves the enzyme and substratein very fast equilibrium with their complex.

6 Transient measuresThis paper is mostly concerned with passage-timemeasurements. Other transient measures can also becalculated via uniformisation. Most generally we cancalculate a probability distribution at a given time from agiven set of source states. Commonly the set of sourcestates may be the singleton set containing only the initialstate of the model. Calculating a series of probabilitydistributions at given time intervals allows the generation ofa time-series plot, which plots the population of a givencomponent type (or state) against time. Our example model

Table 2 Simple Markov chain exhibiting high rates within ameasured passage

0 1 2 3

0 0.1

1 50 000

2 50 000 0.1

3 0.1

Softw., 2009, Vol. 3, Iss. 6, pp. 495–508: 10.1049/iet-sen.2009.0002

of a chemical reaction given in Section 2.3 is analysedusing this more general style of transient analysis. Recallthat the model eventually becomes deadlocked, thusmeaning many other kinds of analysis over the derivedMarkov chain are inappropriate. The resulting time-seriesplot is shown in the graph of Fig. 7. This shows thatinitially the populations of the enzyme (E) and thesubstrate (S) drop rapidly, but after a brief time thepopulation of the enzyme recovers together with anincreasing product (P) population. The opposite happensto the complex (E:S), which initially becomes high inconcentration but decreases as more product is produced.

To compute such a probability distribution we use much ofthe same steps as in passage-time calculations. However, weneed not modify the uniformised chain to contain anabsorbing state since in this case we do wish to calculatethe probability of returning to the source states. Howeverwe must now calculate, for each time point we desire t, foreach hop n the probability that we have performed exactlyn hops at time t.

Because we cannot add an absorbing state, we lose ournovel method of terminating the hop calculations and mustrely on the probability of completing an ever increasingnumber of hops within the time t.

In Section 5 we concluded with the full formulae forcalculating the cdf and pdf of the passage in question. Inthe same spirit of completeness, we provide here the fullformula for calculating the probability distribution of agiven Markov chain at time t from the initial state:

p(t) ¼X1n¼0

(qt)n exp�qt

n!p(n)

� �

Here once again p(n) is the probability distribution as given

Figure 7 Example time-series plot, plotting theconcentration of four species as a function of time fromthe start of the experiment

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by the nth hop value and (qt)n exp�qt =n! is the probabilitythat at time t there have been exactly n hops performed.Note that the sum is from n ¼ 0 since at a given timethere is a possibility that no hops have been performed. Forpassage-time calculations this is not necessary since theprobability of completing the passage within zero hops iszero.

To conclude this section we clear up some terminologywhich we use. A passage-time calculation involvescomputing the probability that the model transitions into atarget state a given time after first entering a source state.Sometimes the set of source states is the singleton setcontaining the initial state of the model. This is commonlyused in ‘time-to-failure’ queries such as: ‘What is theprobability that the server has failed within one week ofoperation?’ When we say transient analysis we usually referto the more general set of transient analyses, which call forthe computation of a probability distribution (even if onlyone state’s probability is examined) a given time after a setof source states have been entered. Such analyses are usedcommonly for biochemical modelling but are also used toanswer such queries as: ‘What is the probabilty that thesystem is operational x minutes after a server crash?’ In thisquery we wish to include the possibility that within the xminutes the server was repaired but crashed again. It istherefore not a passage-time calculation but a more generaltransient analysis.

7 ImplementationThe techniques described in this paper have been fullyimplemented in the International PEPA Compiler (ipc)based on the ipclib [16]. This is a compiler for the PEPA[1], is open source software and may be downloaded from:http://www.dcs.ed.ac.uk/pepa/tools/ipc/.

The input language may be a PEPA model, a Bio-PEPAmodel which may then be converted into a PEPA model or aPetri net specified via the Hydra [3] model file syntax or theXML format used by the PIPE [17] software.

The diagram in Fig. 8 shows the input and outputs of theipc software tool built on top of, and released together withthe ipclib library. The user may start at any level, with allsubsequent levels in the downward direction automaticallyderived from the level above. So the user may start with aBio-PEPA model, which is translated into a PEPA modelor write the PEPA model by hand. In either case they thenprovide a probe specification that is automatically convertedinto a PEPA component and added to the model to give asecond PEPA model and a set of conditions on localcomponent states that define the source and target sets. Orthe user could manually input these source and targetconditions. Alternatively, the user could input a Petri netbut in this case they must specify the source and targetconditions manually since probes are not appropriate for


Petri nets. Whether we now have a PEPA model or a Petrinet, this is compiled into a Markov chain and theconditions are evaluated for all states of the Markov chainto give the source and target sets of states. From this pointall that is left is to derive the performance measure that theuser desires, whether it be a steady-state distribution or atransient measure such as the calculation of passage-timequantiles. Of course, where the users’s query does notinvolve a passage then the target set need not be specified.For a transient measure the source set must still bespecified, although this is commonly the initial state. Forsteady-state distributions and queries that can be derivedfrom it (such as throughput and average response-time),there is no need to even specify the set of source states.

7.1 Comparison with existing techniques

The naive approach that we briefly illustrated in Section 5.1 isto compute for successive values of n until such values dropbelow a threshold. This method may be summed up by

cdf (n, t)¼

0, (p(n)T�Er(n)

t ) , threshold

(p(n)T�Er(n)

t )þ(cdf (nþ 1, t)),

otherwise

8><>:

As we mentioned above this algorithm suffers from aproblem if the input passage has multiple paths tocompletion of varying lengths. In this case the value atsome n may drop below the threshold but may later riseabove the threshold again. The simple solution would stopafter the first time it drops below the threshold.

As an improvement on this technique the Markovianresponse-time analyser Hydra [3, 18–20] monitors thevalue of the erlang distribution with a q rate parameter andn hop parameter. The Hydra solution can therefore be

Figure 8 Inputs and outputs of the International PEPACompiler


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summarised by

cdf (n, t) ¼


(p(n)T� Er(n)

t )þ(cdf (nþ 1, t)),

otherwise

8><>:

Therefore this solution will compute the same values as oursolution in all cases because our solution contains the samecondition. However, our solution is a further refinementthat allows us to avoid needless computation for somevalues of n. In particular, where the t-range – that is thetimes for which we should compute the passage-timequantiles – specified is too large. Suppose the user hasspecified a t-range of 1 – 1000 but the passage has aprobability very close to one of completing by time 500.Because there is a large probability of completing thepassage by time 500 this means that there is a largeprobability of completing the passage within a number ofhops X and that X hops are very likely to be performedwithin 500 time units. This means that for time values over500 there will be a possibility to perform more than X hopsand the Hydra solution will continue to computeprobabilities for these hop values. However, our solutionwould recognise that such values cannot add anything tothe cdf because you are very like to have completed thepassage before X hops. In the case of the cdf this could bemitigated by incorporating the naive solution but this is notas effective as for computing the pdf.

Our solution has a further, related, advantage; the userneed not specify the upper bound on the t-range at all. Theuser need only give the start of the t-range and the steps inwhich we wish to increase the value of t. This is becauseusing our technique we can calculate the value X at whichperforming more than X number of hops will notsignificantly add to the probability of completing thepassage (because there is a probabilty within the thresholdof being in the absorbing state by X hops). We can thenuse this to work out the upper bound on the t-range bycalculating the value of t such that performing X hopswithin time t is significantly likely. In order that the userneed not specify a t-range at all we default to a startingtime of zero and a time step of the calculated stop-timedivided by one hundred. The user may then override any ofthe start time, the stop time, the time increments and thenumber to divide the t-range by in order to obtain the timeincrements.

8 ConclusionsIn this paper we have detailed a software library ipclib andassociated tool ipc for the derivation of performancemeasures from process algebra models that may betranslated into continous time Markov chains. Once theMarkov chain has been derived the common steady-stateprobability distribution and associated performancemeasures such as throughput and average passage duration


may be computed. In this paper, though, we have focusedon the transient measure facilities provided by ipc. Inparticular, the calculation of passage-time quantiles isprovided for via the compositional approach to model andquery specification. In particular, the stochastic probesspecification language allows the compiler to automaticallyprepare the Markov chain in a suitable way to avoid aninvalid passage-time computation. It is very encouragingthat the constraints on a valid passage-time calculation arenot only checked but where there is a violation automaticstate duplication occurs in order to amend the passage intoa suitable form. Moreover, states not required within theuniformised Markov chain may be readily determined andthus removed in order to optimise the query.

We have also given a detailed account of the calculation ofpassage-time quantiles and densities from continuous-timeMarkov chains. We have shown the process from non-uniformised Markov chain through the uniformisation,modification with an absorbing state, the calculation of thehop probability distributions and the calculations involvedin producing the final cdf and pdf. Two importantproperties that allow the otherwise infinite calculation toterminate were identified which have the desiredconservative feature – meaning that we never terminate tooearly producing an erroneous answer. However, thecombination of the two provides early detection to avoidsome needless calculations. Finally, we feel that our paperprovides a good introduction to the topic of uniformisationin general.

9 AcknowledgmentsThe authors are supported by the EU FET-IST GlobalComputing 2 project SENSORIA (Software Engineeringfor Service-Oriented Overlay Computers (IST-3-016004-IP-09)). The ipc/Hydra tool chain has been developed inco-operation with Jeremy Bradley, Will Knottenbelt andNick Dingle of Imperial College, London.

10 References

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[2] CIOCCHETTA F., HILLSTON J.: ‘Bio-PEPA: a framework for themodelling and analysis of biological systems’, Theoreti.Comput. Sci., 2008, 410, (33-34), pp. 3065–3084

[3] DINGLE N.J., KNOTTENBELT W.J., HARRISON P.G.: ‘HYDRA:HYpergraphbased Distributed Response-time Analyser’.Proc. 2003 Int. Conf. on Parallel and DistributedProcessing Techniques and Applications, 2003, vol. 1,pp. 215–219

[4] CLARK G., COURTNEY T., DALY D., DEAVOURS D., DERISAVI S., DOYLE

J.M., SANDERS W.H., WEBSTER P.: ‘The Mobius modeling tool’.

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PNPM’01: Proc. Ninth Int. Workshop on Petri Nets andPerformance Models (PNPM’01), USA, 2001, p. 241

[5] TRIBASTONE M., DUGUID A., GILMORE S.: ‘The PEPA EclipsePlug-in’, Perform. Eval. Rev., 2009, 36, (4), pp. 28–33

[6] DIENER J.D.: ‘Empirical comparison of uniformizationmethods for continuous-time Markov chains’, inSTEWART W.J. (ED.): ‘Computations with Markov chains’(Kluwer Academic Publishers, 1995), pp. 547–570

[7] KATOEN J.P., KHATTRI M., ZAPREEV I.S.: ‘A Markov rewardmodel checker’. Quantitative Evaluation of Systems(QEST), 2005, pp. 243–244

[8] KATOEN J.P., ZAPREEV I.S.: ‘Safe on-the-fly steady-statedetection for timebounded reachability’. QuantitativeEvaluation of Systems (QEST), 2006, pp. 301–310

[9] KWIATKOWSKA M., NORMAN G., PARKER D.: ‘PRISM 2.0: a toolfor probabilistic model checking’. Proc. First Int. Conf. onQuantitative Evaluation of Systems (QEST’04), 2004,pp. 322–323

[10] AZIZ A., SANWAL K., SINGHAL V., BRAYTON R.: ‘Model-checkingcontinuoustime Markov chains’, ACM Trans. Comput.Logic, 2000, 1, (1), pp. 162–170

[11] CALDER M., GILMORE S., HILLSTON J.: ‘Modelling the influenceof RKIP on the ERK signalling pathway using the stochasticprocess algebra PEPA’. Transactions on ComputationalSystems Biology VII, 2006, (LNCS, 4230)

[12] CALDER M., GILMORE S., HILLSTON J.: ‘Automatically derivingODEs from process algebra models of signallingpathways’. Proc. Computational Methods in SystemsBiology (CMSB 2005), 2005, pp. 204–215


[13] CLARK A., GILMORE S.: ‘State-aware performance analysiswith eXtended Stochastic Probes’. Proc. Fifth EuropeanPerformance Engineering Workshop (EPEW 2008), 2008,(LNCS, 5261), pp. 125–140

[14] ARGENT-KATWALA A., BRADLEY J., CLARK A., GILMORE S.: ‘Location-aware quality of service measurements for service-levelagreements’. Proc. Third Int. Conf. on Trustworthy GlobalComputing (TGC’07), 2008, (LNCS, 4912), pp. 222–239

[15] BAIER C., HAVERKORT B.R., HERMANNS H., KATOEN J.P.: ‘Modelchecking continuous-time Markov chains by transientanalysis’. CAV’00: Proc. 12th Int. Conf. on Computer AidedVerification, 2000, pp. 358–372

[16] CLARK A.: ‘The ipclib PEPA Library’. Proc. Fourth Int.Conf. Quantitative Evaluation of SysTems (QEST), 2007,pp. 55–56

[17] DINGLE N.J., KNOTTENBELT W.J., SUTO T.: ‘PIPE2: a tool for theperformance evaluation of generalised stochastic PetriNets’, SIGMETRICS Perform. Eval. Rev., 2009, 36, (4),pp. 34–39

[18] KNOTTENBELT W.J.: ‘Generalised Markovian analysis oftimed transition systems’. Master’s thesis, Department ofComputer Science, University of Cape Town, 1996

[19] DINGLE N.J.: ‘Parallel computation of responsetime densities and quantiles in large markov and semi-Markov models’. PhD thesis, Department of Computing,Imperial College London, University of London, 2004

[20] BRADLEY J., DINGLE N., GILMORE S., KNOTTENBELT W.: ‘Extractingpassage times from PEPA models with the HYDRA tool:a case study’. Proc. 19th Annual UK Performance EngineeringWorkshop, 2003, University of Warwick, pp. 79–90


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