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Journal ArticleDOI

Performance Evaluation of Distributed Systems Based on a Discrete Real- and Stochastic-Time Process Algebra

01 Jan 2009-Fundamenta Informaticae (IOS Press)-Vol. 95, Iss: 1, pp 157-186
TL;DR: Previous investigations on the topic, which only touched long-run analysis, are extended to tackle transient analysis as well and the theoretical results obtained allow us to extend the χ-toolset.
Abstract: We present a process-algebraic framework for performance evaluation of discrete-time discrete-event systems. The modeling of the system builds on a process algebra with conditionallydistributed discrete-time delays and generally-distributed stochastic delays. In the general case, the performance analysis is done with the toolset of the modeling language χ by means of discrete-event simulation. The process-algebraic setting allows for expansion laws for the parallel composition and the maximal progress operator, so one can directly manipulate the process terms and transform the specification in a required form. This approach is illustrated by specifying and solving the recursive specification of the G/G/1/∞ queue, as well as by specifying a variant of the concurrent alternating bit protocol with generally-distributed unreliable channels. In a specific situation when all delays are assumed deterministic, we turn to performance analysis of probabilistic timed systems. This work employs discrete-time probabilistic reward graphs, which comprise deterministic delays and immediate probabilistic choices. Here, we extend previous investigations on the topic, which only touched long-run analysis, to tackle transient analysis as well. The theoretical results obtained allow us to extend the χ-toolset. For illustrative purposes, we analyze the concurrent alternating bit protocol in the extended environment of the χ-toolset using discrete-event simulation for generallydistributed channels, the developed analytical method for deterministic channels, and Markovian analysis for exponentially-distributed delays.

Summary (2 min read)

1. Introduction

  • Over the past decade stochastic process algebras have emerged as compo iti nal modeling formalisms for systems that not only require functional verification, but performance analysis as well.
  • Similarly, the semantics of stochastic process algebras is given using clocks that represent the stochastic delays at the symbolic level.
  • For the sampling of the clock two execution policies can be adopted: (1) racecondition [26, 20, 31, 10], which enables the action transitions guarded by the clocks that expirefirst, and (2) pre-selection policy [13, 12], which preselects the clocks by a probabilistic choice.
  • The algebra also providesthe possibility of specifying a partial race of stochastic delays, e.g., that one delay has always a shorter, equal, or longer sample than the other delay.
  • For analysis of the concurrent alternating bit protocol the authors depend on the toolse of theχ-language [8, 38, 11, 2].

2. Timed and Stochastic Delays

  • The authors refer the interested reader for more technical detail to [32].
  • Therefore, the names of the losing delays must be protected inp, i.e., they become dependent.
  • Byσ∅ X , the authors denote the event where the delay does not expire in one time unit, i.e., the stochastic delayX loses the race to a unit time delay and there are no additional winners.
  • Also, the authors favor weak choice between immediate actions and passage of time, i.e., they impose a nondeterministic choice on the immediate actions andthe passage of time in the vein of the timed process algebras of [4].

3. Process Theory

  • In this section the authors introduce the process theoryTCPdst of communicating processes with discrete real and stochastic time for race-complete process specifications that induce races with all possible outcomes.
  • The general idea of having both dependent and independent delays available is the following:.
  • To denote that after a delay[WL ], the same time that passed for the winnersW has also passed for the losersL, the authors use an environmentα : V → N. For eachX ∈ V, α(X) represents the amount of time thatX has raced.
  • Axiom A17 states that if the losers of the first timed delay have acommon delay with the winners of the second, then all delays of the second delay are losers in the resulting delay.
  • Finally, the axioms A19–A21 give the standard axioms for the encapsulation operator that suppresses the actions inH.

4. Performance Evaluation

  • For the purpose of performance analysis, the authors choose the framework ofthe languageχ.
  • This provides for a better expressivity and modeling convenience [33].
  • The discrete-time probabilistic reward graph is represented as an equivalent discrete-time Markov reward chain, which is then analyzed, and the results are interpreted back in the discrete-time probabilistic reward graph setting.
  • The aggregation eliminates the probabilistic states4 and5 and splits the incoming timed transitions from the states6 and3.
  • The multiplication of the transition matrix ofM with its folding collector produces the accumulative probability of residing in each unfolded timed state ofM per unfolding set.

5. The Concurrent Alternating Bit Protocol

  • The authors specify the concurrent alternating bit protocol both in the process theoryTCPdst and in the specification languageχ.
  • If the acknowledgement is received before the timeout expires, the process flips the alternating bit, packs the new data intp time units, and sends it again via channelc3.
  • The authors takea similar approach as for the absence of a probabilistic choice, and add rewards by manipulating theχ specification (again side-stepping changes inχ), see Figure 6 below.
  • Figure 9 gives the utilization of the data channelK, when the distribution of the delay of the data channel is uniform between2 and10 and the distribution of the delay of the acknowledgement channel is uniform between1 and4.
  • The Markovian analysis always underestimates the performance because the expected value of the maximum of two exponential d lays is greater than maximum of the expected values of both delays.

6. Conclusion

  • The authors proposed a performance evaluation framework that is based on a process theory that enables specification of distributed systems with discrete timed and stochastic delays.
  • The authors provided expansion laws for the parallel composition and the maximal progress operat r.
  • The authors gave transient analysis of these models by translating them to discrete-time Markov reward chains.
  • This should pave the way for bigger case studies on Internet protocol verification and analysis as detailed performance specification becomes viable by using both generally-distributed stochastic delays and standard timeou s.
  • Many thanks to Jos Baeten for fruitful discussions on the topic.

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Fundamenta Informaticae XX (2009) 1–29 1
IOS Press
Performance Evaluation of Distributed Systems Based on
a Discrete Real- and Stochastic-Time Process Algebra
J. Markovski and E.P. de Vink
Formal Methods Group, Department of Mathematics and Computer Science
Eindhoven University of Technology, Den Dolech 2, 5612 AZ Eindhoven, The Netherlands
tel: +31 40 247 3360, fax: +31 40 247 5361
j.markovski@tue.nl, evink@win.tue.nl
Abstract. We present a process-algebraic framework for performance evaluation of discrete-time
discrete-event systems. The modeling of the system builds on a process algebra with conditionally-
distributed discrete-time delays and generally-distributed stochastic delays. In the general case, the
performance analysis is done with the toolset of the modeling language χ by means of discrete-event
simulation. The process-algebraic setting allows for expansion laws for the parallel composition and
the maximal progress operator, so one can directly manipulate the process terms and transform
the specification in a required form. This approach is illustrated by specifying and solving the
recursive specification of the G/G/1/ queue, as well as by specifying a variant of the concurrent
alternating bit protocol with generally-distributed unreliable channels. In a specific situation when
all delays are assumed deterministic, we turn to performance analysis of probabilistic timed systems.
This work employs discrete-time probabilistic reward graphs, which comprise deterministic delays
and immediate probabilistic choices. Here, we extend previous investigations on the topic, which
only touched long-run analysis, to tackle transient analysis as well. The theoretical results obtained
allow us to extend the χ-toolset. For illustrative purposes, we analyze the concurrent alternating bit
protocol in the extended environment of the χ-toolset using discrete-event simulation for generally-
distributed channels, the developed analytical method for deterministic channels, and Markovian
analysis for exponentially-distributed delays.
1. Introduction
Over the past decade stochastic process algebras have emerged as compositional modeling formalisms
for systems that not only require functional verification, but performance analysis as well. Many Marko-
vian process algebras are developed like EMPA [9], PEPA [27], IMC [25], etc. exploiting the memoryless
Address for correspondence: J. Markovski, TU/e, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

2 J. Markovski, E.P. de Vink / A Discrete-Time Process Algebraic Framework for Performance Evaluation
property of the exponential distribution. Before long, the need for general distributions arose, as expo-
nential delays are not sufficient to model, for example, fixed timeouts of Internet protocols or heavy-tail
distributions present in media streaming services. Prominent stochastic process algebras and calculi with
general distributions include TIPP [26], GSMPA [13], SPADES [20], IGSMP [12], NMSPA [31], and
MODEST [10].
Despite the greater expressiveness, compositional modeling with general distributions proved to be
challenging, as the memoryless property cannot be relied on [29, 14]. Typically, the underlying perfor-
mance model is a generalized semi-Markov process that exploits clocks to memorize past behavior in
order to retain the Markov property of history independence [23]. Similarly, the semantics of stochastic
process algebras is given using clocks that represent the stochastic delays at the symbolic level. Such a
symbolic representation allows for the manipulation of finite structures, e.g., stochastic automata or ex-
tensions of generalized semi-Markov processes. The concrete execution model is subsequently obtained
by sampling the clocks, frequently yielding infinite probabilistic timed transition systems.
For the sampling of the clock two execution policies can be adopted: (1) race condition [26, 20, 31,
10], which enables the action transitions guarded by the clocks that expire first, and (2) pre-selection
policy [13, 12], which preselects the clocks by a probabilistic choice. To keep track of past behavior,
the clock samples have to be updated after each stochastic delay transition. One can do this in two
equivalent ways: (1) by keeping track of residual lifetimes [20, 10], i.e., the time left up to expiration,
or (2) by keeping track of the spent lifetimes [26, 13, 12, 31], i.e., the time passed since activation. The
former manner is more suitable for discrete-event simulation, whereas the latter is acknowledged for its
correspondence to real-time semantics [29, 14].
In this paper we consider the race condition with spent-lifetime semantics. However, we do not use
clocks to implement the race condition and to determine the winning stochastic delay(s) of the race.
Rather, we rely on an interpretation that uses conditional random variables and makes a probabilistic
assumption on the winners followed by conditioning of the distributions of the losers on the time spent
for the winning samples [28]. Thus, we no longer speak of clocks as we do not keep track of sample
lifetimes, but we only cater for the ages of the conditional distributions [35]. We refer to the samples as
stochastic delays, a naming resembling standard timed delays.
The relation between real and stochastic time has been studied in various settings: a structural trans-
lation from stochastic to timed automata with deadlines is given in [19]. This approach found its way
into MODEST, where timed automata with deadlines are merged with stochastic automata in so-called
stochastic timed automata as a means to introduce real and stochastic time as separate constructs. Also, a
translation from IGSMP into pure real-time models called interactive timed automata is reported in [12].
The interplay between standard timed delays and discrete stochastic delays has been studied in [34, 35].
An axiomatization for a process algebra that embeds real-time delays with so-called context-sensitive
interpolation into a restricted form of discrete stochastic time is given in [35].
The paper presents a performance evaluation framework based on process algebraic specifications
and their analysis in an extended environment of the χ-toolset [8, 38]. The contribution of the paper is
twofold. As a first contribution, a sound and ground-complete process algebra is provided that accom-
modates timed delays in a racing context, extending the work of [34, 35]. The theory provides an explicit
maximal progress operator and a non-trivial expansion law for the parallel composition. Differently
from other approaches, we derive stochastic delays as time-delayed processes with explicit information
about the winners and the losers that induced the delay. We represent standard real-time as stochastic
time inducing a trivial race condition in which the shortest sample is always exhibited by the same set

J. Markovski, E.P. de Vink / A Discrete-Time Process Algebraic Framework for Performance Evaluation 3
of delays and moreover has a fixed duration. The algebra also provides the possibility of specifying a
partial race of stochastic delays, e.g., that one delay has always a shorter, equal, or longer sample than
the other delay. This is required when modeling timed systems whose correct behavior depends on the
relative ordering of the timed delays, e.g., in a time dependent controller. When the timed delays are
simply replaced by stochastic delays, the total order of the samples is, in general, lost, unless it can be
specified which delays are the winners or losers of the imposed race.
We illustrate the process theory by revisiting the G/G/1/ queue from [34], treating it more ele-
gantly now and providing a solution for the recursive specification by manipulating process terms using
the proposed axiomatization. We also specify a variant of the concurrent alternating bit protocol that has
fixed timeouts (represented by timed delays) and faulty generally-distributed channels (represented by
stochastic delays), stressing the interplay of real-time and stochastic time.
Our second contribution concerns automated performance analysis. It is well known that only a small
number of restricted classes of models of general distributions are analytically solvable. Preliminary
research on model checking of stochastic automata is reported in [15] and a proposal for model checking
probabilistic timed systems is given in [39]. However, at the moment, performance analysts turn to
discrete-event simulation when it comes to analyzing models with generally-distributed delays. For
analysis of the concurrent alternating bit protocol we depend on the toolset of the χ-language [8, 38, 11,
2]. At the start, χ was used to model discrete-event systems only, not supported by an explicit semantics.
However, recently, it has been turned into a formal specification language set up as a hybrid process
algebra with data [8, 38].
The connection between the timed discrete-event subset of χ and standard timed process algebras
in vein of [4] is straightforward. In [42], a proposal was given to extend χ with a probabilistic choice
to enable long-run performance analysis of probabilistic timed specifications. Here, we rely on this ex-
tension to provide a connection with the stochastic part of our process algebra as well. At this point,
the co-existence of real and stochastic time in the same model plays a crucial role, which underlines the
key position of the process algebra in the framework. The performance model is termed discrete-time
probabilistic reward graph and it comprises deterministic delays and immediate probabilistic choices. It
is suitable as an underlying performance model for stochastic delays with finite support set as used in the
case study (even though the theory does not have such a limitation). In [42], discrete-time probabilistic
reward graphs were employed for long-run analysis of industrial systems. Here, we extend the perfor-
mance evaluation framework of [42] to cater for transient analysis as well. We accordingly augment the
χ-toolset and apply it to the concurrent alternating bit protocol. The case study illustrates the new ap-
proach when the channel distributions are deterministic. Finally, we compare the analytical results with
the ones obtained from discrete-event simulation and Markovian analysis using the same specification
in χ. We visualize the proposed framework in Figure 1. We note that we rely on the CADP toolset [21]
as a solver for the underlying/intermediate Markov reward processes.
The rest of this paper is organized as follows: Section 2 discusses background material and design
choices. Section 3 introduces the process theory and revisits the G/G/1/ queue example. Section 4
discusses transient analysis of discrete-time probabilistic reward graph in the performance evaluation
framework. Section 5 analyzes the concurrent alternating bit protocol protocol and discusses its specifi-
cation in the proposed process algebra and the language χ. Section 6 wraps up with concluding remarks.
Due to substantial technical overhead, we do not give the operational semantics of the process-algebraic
theory here. Instead, we focus on the axiomatization to illustrate its suitability for protocol specification.
The complete structural operational semantics and formal treatment of the theory are available in [32].

4 J. Markovski, E.P. de Vink / A Discrete-Time Process Algebraic Framework for Performance Evaluation
Manipulation of processes with
discrete timed and generally-
distributed stochastic delays:
Process algebra TCP
dst
Performance evaluation of
generally-distributed
processes:
Chi-simulator
Performance evaluation of
probabilistic timed processes:
Timed Chi to Discrete-time
probabilistic reward graphs +
CADP toolset
Performance evaluation of
geometrically/ exponentially-
distributed processes:
Markovian extension of Chi
Figure 1. The proposed process-algebraic performance evaluation framework
2. Timed and Stochastic Delays
In this section we introduce a number of notions in process theory that are used below. We refer the
interested reader for more technical detail to [32].
Preliminaries We use discrete random variables to represent durations of stochastic delays. The
set of discrete distribution functions F such that F(n)=0 for n 0 is denoted by F; the set of the
corresponding random variables by V. We use X, Y , and Z to range over V and F
X
, F
Y
and F
Z
for
their respective distribution functions. Also, W , L, V , and D range over 2
V
. Given a set A, by A
n
we
denote vectors of size n N and by A
m×n
matrices with m rows and n columns with elements in A.
By 0 and 1 we denote vectors that consist of 0s and 1s.
Racing stochastic delays A stochastic delay is a timed delay of a duration guided by a random
variable. We observe simultaneous passage of time for a number of stochastic delays until one or some
of them expire. This phenomenon is referred to as the race condition and the setting as the race. For
multiple racing stochastic delays, different stochastic delays may be observed simultaneously as being
the shortest. The ones that have the shortest duration are called the winners and the others are referred
to as the losers. The outcome of a race is completely determined by the winners and the losers and their
distributions. So, we can explicitly represent the outcome of the race by a pair of sets W, L of stochastic
delays. We write [
W
L
] in case W is the set of winners and L is the set of losers. We have occasion to
write [W ] instead of [
W
] and omit the set brackets when clear from the context. Thus, [X] represents a
stochastic delay guided by the random variable X.
To express a race, we will use the operator
+ . So, [X] + [Y ] represents the race between the
stochastic delays X and Y . There are three possible outcomes of this race: (1) [
X
Y
], (2) [
X, Y
], and (3) [
Y
X
].
Thus, we can also write [
X
Y
] + [
X, Y
] + [
Y
X
] instead of [X] + [Y ], as both expressions represent the same
final outcomes of a race. If an additional racing delay Z is added, this also leads to equal outcomes, i.e.,
[X] + [Y ] + [Z] and [
X
Y
] + [
X, Y
] + [
Y
X
] + [Z] will yield the same behaviour. For example, the outcome of
[
X
Y
] + [Z] is either (1) [
Z
X, Y
], (2) [
X, Z
Y
], or (3) [
X
Y, Z
]. As outcomes of races may be involved in other races,
we generalize the notion of a stochastic delay and refer to an arbitrary outcome [
W
L
] as a stochastic delay
induced by the winners W and the losers L, or by W and L for short. Here, we decide not to dwell on

J. Markovski, E.P. de Vink / A Discrete-Time Process Algebraic Framework for Performance Evaluation 5
the formal semantics because of a substantial technical overhead to formalize the notion of dependencies
of losers on the samples of the winners. The basis for the semantics is given in [34, 35] and subsequently
extended in [32] to allow the explicit specification of the winners and the losers of a race.
To summarize, there are three possible combinations that give the relation between the winners and
the losers: (1) L
1
W
2
6= , which means that the race must be won by W
1
and lost by L
1
W
2
L
2
,
(2) W
1
W
2
6= , which means that the race must be won by W
1
W
2
together and lost by L
1
L
2
,
and (3) W
1
L
2
6= , which means that the race must be won W
2
and lost by W
1
L
1
L
2
. Obviously,
these ‘restrictions’ are disjoint and cannot be applied together. If more than one restriction holds, then
they lead to ill-defined outcomes. For example, if both (1) and (2) hold at the same time, then L
1
and W
2
must exhibit the same sample and also W
1
and W
2
must exhibit the same sample. Then W
1
and L
1
must
exhibit the same sample, which is a contradiction.
If at least two restrictions apply, then the outcomes cannot be combined as they represent disjoint
events. In this case we say the race between the delays [
W
1
L
1
] and [
W
2
L
2
] with W
1
L
1
= W
2
L
2
, is
resolved. The extra condition ensures that the outcomes stem from the same race, i.e, they have the same
racing delays. For example, [
X
Y
] and [
Y, Z
X
] cannot form a joint outcome. The delays do not stem from the
same race, which renders their combination inconsistent. Resolved races play an important role as they
enumerate every possible outcome of the race. We define a predicate rr([
W
1
L
1
], [
W
2
L
2
]) that checks whether
two delays [
W
1
L
1
] and [
W
2
L
2
] are in a resolved race. It is satisfied if W
1
L
1
= W
2
L
2
and at least two of the
following three restrictions from above hold: (1) L
1
W
2
6= , (2) W
1
W
2
6= , and (3) W
1
L
2
6= .
Naming of stochastic delays Consider the process term [X].p
1
k[X].p
2
, where [X].
denotes stochastic
delay prefixing,
k denotes the parallel composition, and p
1
and p
2
are arbitrary process terms. We note
that the alternative and the parallel composition impose the same race condition. In a standard way, the
race is performed on two stochastic delays with the same distribution F
X
F. However, both delays
will not necessarily exhibit the same sample, unless F
X
is Dirac. Intuitively, the process given by the
above term is equivalent to process given by [X].p
1
k [Y ].p
2
with F
X
= F
Y
leading to three possible
outcomes.
However, in real-time semantics, timed delays (denoted by σ
n
for a duration n N) with the same
duration are merged together. For example, σ
m
.p
1
k σ
m
.p
2
is equivalent to σ
m
.(p
1
k p
2
). This parallel
composition represents components that should delay together. Note that this is not obtained above in
the stochastic setting. Previous investigation in this matter [34, 35, 32] points out that both dependent
and independent stochastic delays are indispensable. The former enable an expansion law for the parallel
composition; the latter support compositional modeling.
Dependent stochastic delays always exhibit the same duration in the same race when guided by the
same random variable. In contrast, independent stochastic delays with the same name have the same
distribution, but not necessarily the same duration. As an example, [
X, Y
Z
] + [
X
U
] is the same race as [
X, Y
Z, U
]
if we treat X as a dependent stochastic delay, whereas [
X
Z
] + [X] = [
X
Z, Y
] + [
X, Y
Z
] + [
Y
X, Z
], provided that
F
X
= F
Y
, when X is treated as an independent one.
We introduce an operator to specify dependent delays, denoted by |
|
D
, in which scope the stochastic
delays in D are treated as dependent. Thus, in the previous example, |[
X, Y
Z
]|
X
denotes that X is a
dependent stochastic delay, but Y and Z are independent. By default, every delay is considered as
dependent. Hence, [
W
L
] actually means |[
W
L
]|
W L
. Multiple scope operators intersect and, e.g., ||[
X
Y
]|
X
|
Y
denotes the independent delay [
X
Y
] because {X} {Y } = .
The dependence scope plays an important role in giving operational semantics to the terms. Recall

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Journal ArticleDOI
TL;DR: In this paper, a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond is presented, which is a book of applicable mathematics, which avoids the approach of listing only the techniques, followed by a few examples.
Abstract: This is a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond. It is a book of applicable mathematics. It avoids the approach of listing only the techniques, followed by a few examples, without explaining why the techniques work. Thus, it provides not only the know-how but also the know-why. Equally, the text has not been written as a book of pure mathematics with a list of theorems followed by their proofs. The authors' aim is to help students develop an understanding of mathematics and its applications. They have refrained from using clichés like “it is obvious” and “it can be shown”, which may be true only to a mature mathematician. On the whole, the authors have been generous in writing down all the steps in solving the example problems.The book comprises ten chapters. Each chapter contains several solved problems clarifying the introduced concepts. Some of the examples are taken from the recent literature and serve to illustrate the applications in various fields of engineering and science. At the end of each chapter, there are assignment problems with two levels of difficulty. A list of references is provided at the end of the book.This book is the product of a close collaboration between two mathematicians and an engineer. The engineer has been helpful in pinpointing the problems which engineering students encounter in books written by mathematicians.

2,846 citations

Book
13 Jun 1996
TL;DR: Modelling study: multi-server multi-queue systems shows strong equivalence between strong and weak isomorphism and strong bisimilarity.
Abstract: 1. Introduction 2. Background 3. Performance evaluation process algebra 4. Modelling study: multi-server multi-queue systems 5. Notions of equivalence 6. Isomorphism and weak isomorphism 7. Strong bisimilarity 8. Strong equivalence 9. Conclusions Bibliography Index.

1,187 citations


"Performance Evaluation of Distribut..." refers methods in this paper

  • ...Many Markovian process algebras are developed like EMPA [9], PEPA [27], IMC [25], etc. exploiting the memoryless Address for correspondence: J. Markovski, TU/e, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands property of the exponential distribution....

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Book
11 Aug 1971

983 citations

Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "Performance evaluation of distributed systems based on a discrete real- and stochastic-time process algebra" ?

The authors present a process-algebraic framework for performance evaluation of discrete-time discrete-event systems. In a specific situation when all delays are assumed deterministic, the authors turn to performance analysis of probabilistic timed systems. This work employs discrete-time probabilistic reward graphs, which comprise deterministic delays and immediate probabilistic choices. For illustrative purposes, the authors analyze the concurrent alternating bit protocol in the extended environment of the χ-toolset using discrete-event simulation for generallydistributed channels, the developed analytical method for deterministic channels, and Markovian analysis for exponentially-distributed delays. 

As future work, the authors plan to introduce the hiding operator that produces internal transitions and to develop a notion of branching or weak bisimulation in that setting. The authors can also exploit existing real-time specification as the theory is sufficiently flexible to allow extension of real-time with stochastic time while retaining any imposed ordering of the original delays. The authors are indebted to the reviewers for their constructive comments and suggestions.