The purpose of this paper is to demonstrate the exibility of the SPA

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1 TIPP and the Spectral Expansion Method I. Mitrani 1, A. Ost 2, and M. Rettelbach 2 1 University of Newcastle upon Tyne 2 University of Erlangen-Nurnberg Summary. Stochastic Process Algebras (SPA) like TIPP are a means for functional, performance and dependability modelling of concurrent systems in a modular fashion. Until now their applicability has been restricted by the requirement that the process state space should be nite. This was due to the solution algorithms that were employed. In this paper we present a variant of SPA which enables the Spectral Expansion solution method (SE) to be used, thus allowing the modelling of processes with innite state space. Whether SE is applicable to a given problem can be decided before generating a detailed description of the state space. The mapping from the SPA to the SE formalisms can be automated and the technicalities of the solution can be hidden from the user. This approach is illustrated on a small but non-trivial example. 1. Introduction Stochastic Process Algebras provide an \easy-to-use" interface for the description of stochastic processes. They support modular design in a natural way by providing operators for the construction of complex systems from smaller ones. A particular SPA called TIPP, dealing with Markov processes only, was introduced in [2]. In common with other SPA's, the current version of TIPP requires that the system to be modelled has a nite number of states. The TIPP description is transformed into a nite Markov chain; the latter is then solved numerically in order to determine the steady-state probabilities. The purpose of this paper is to demonstrate the exibility of the SPA approach by showing how a small modication of the syntax, combined with appropriate new semantics, can extend the modelling power considerably. The target Markov process can now be two-dimensional, nite in one direction and innite in the other. The model description in terms of the modied TIPP (called SE-TIPP) is transformed into a collection of matrices which are then fed into the Spectral Expansion solution procedure [6]. The innite Markov chain is solved exactly by exploiting the inherent regularity. The semantic concepts introduced in this paper may appear complex but it must be emphasized that most of the technical details are hidden for a normal user. Thus one can model a system with the SPA formalism (following a few simple rules) and apply SE to the problem. The matrices necessary for the SE solution are generated automatically. The models that are being considered are dened in section 2, which also provides a reasonably self-contained introduction to the SE solution method. Section 3 describes the modied Stochastic Process Language SE-TIPP and denes the special semantic rules for generating the SE matrices. In Section 4,

2 2 I. Mitrani et al. the schema is applied to an example taken from the area of ATM/B-ISDN networks. 2. The Spectral Expansion solution method There is a large class of models which involve two-dimensional Markov processes on semi-innite lattice strips. That is, the system state is described by two integer random variables, I and J ; one of these has a nite range, and the other can take any non-negative value. Often these models are cast in the framework of a Markov-modulated queue: then the bounded variable, I, indicates the state of the Markovian environment, while the unbounded one, J, represents the number of jobs in the system (a recent survey can be found in Prabhu and Zhu [8]). We are interested in a sub-class of the above processes, characterised by the following two properties: (i) the instantaneous transition rates out of state (i; j) do not depend on j when the latter is suciently large; (ii) the jumps of the random variable J are limited in size. To keep the presentation simple, we shall describe the Spectral Expansion solution method in the context of a skip-free Markov-modulated queue (i.e., the jobs arrive and depart singly). For a more general treatment, and a comparison with other solution methods, see [1, 6]. Suppose that the Markovian environment can be in N possible states, numbered ; 1; : : :; N? 1. Let I(t) and J(t) be the random variables representing the state of the environment at time t, and the number of jobs in the system at time t, respectively. We shall sometimes refer to I(t) as the operative state. It is assumed that X = f[i(t); J(t)] ; t g is an irreducible Markov process with state space f; 1; : : :; N? 1g f; 1; : : :g. The evolution of that process proceeds according to the following set of possible transitions: (a) From state (i; j) to state (k; j) ( i; k N? 1 ; i 6= k ); (b) From state (i; j) to state (k; j + 1) ( i; k N? 1 ); (c) From state (i; j) to state (k; j? 1) ( i; k N? 1 ). We assume further that there is a threshold, M, (M 1 ) such that the instantaneous transition rates do not depend on j when j M. In other words, if we denote the transition rate matrices associated with (a), (b) and (c) by A () j, A (1) j and A (?1) j respectively (the main diagonal of A () j is zero by denition; also, A (?1) = by denition), then we have A () j = A () M ; A(1) j = A (1) M ; A(?1) j = A (?1) M ; j M : (2.1) In the above notation, the superscript denotes the jump in the number of jobs present. Transitions (a) correspond to changes in the environment only.

3 TIPP and the Spectral Expansion Method 3 A transition of type (b) represents a job arrival which may coincide with such a change. If these coincidences do not occur, then the matrices A (1) j and A (1) M are diagonal. Similarly, a transition of type (c) represents a job departure coinciding with a change in the environment. Again, if such coincidences do not occur, then the matrices A (?1) j and A (?1) M are diagonal. As well as these matrices, it is convenient to dene the diagonal matrices D () j, D (1) j and D (?1) j, whose i th diagonal element is the i th row sum of A () j, A (1) j and A (?1) j, respectively. Those row sums are the total rates at which the process X leaves state (i; j), due to (a) changes in the environment, (b) job arrivals (perhaps accompanied by such a change) and (c) job departures (ditto), respectively. The j -independent versions of these diagonal matrices are denoted by D () M, D(1) M and D(?1) M, respectively. The object of the analysis is to determine the joint steady-state distribution of the state of the environment and the number of jobs in the system: p i;j = lim t!1 P (I(t) = i; J(t) = j) ; i = ; 1; : : :; N? 1 ; j = ; 1; : : : :(2.2) That distribution exists for an irreducible Markov process if, and only if, the corresponding set of balance equations has a unique normalisable solution. Rather than working with the two-dimensional distribution fp i;j g, we shall introduce the row vectors, v j = (p ;j ; p 1;j ; : : :; p N?1;j ) ; j = ; 1; : : : ; (2.3) whose elements represent the states with j jobs in the system. The balance equations satised by the probabilities p i;j can be written in terms of the vectors v j : v j [D () j + D (1) j + D (?1) j ] = v j?1 A (1) j?1 +v ja () j +v j+1 A (?1) j+1 ; j M ;(2.4) (where v?1 = by denition), and v j [D () M + D(1) M + D(?1) M ] = v j?1a (1) M + v ja () M + v j+1a (?1) M ; j > M :(2.5) In addition, all probabilities must sum up to 1: 1X j= v j e = 1 ; (2.6) where e is a column vector with N elements, all of which are equal to 1. The rst step is to nd the general solution of equation (2.5). That equation has the nice property that its coecients do not depend on j. It can be rewritten in the form v j Q + v j+1 Q 1 + v j+2 Q 2 = ; j = M; M + 1; : : : ; (2.7) where Q = A (1) M, Q 1 = A () M? D() M? D(1) M? D(?1) M and Q 2 = A (?1) M. This is a homogeneous vector dierence equation of order 2, with constant coecients. Associated with it is the characteristic matrix polynomial, Q(), dened as

4 4 I. Mitrani et al. Q() = Q + Q 1 + Q 2 2 : (2.8) Denote by k and k the eigenvalues and corresponding left eigenvectors of Q(). In other words, these are quantities which satisfy kq( k ) = ; k = 1; 2; : : :; d ; (2.9) where d = degreefdet[q()]g. The eigenvalues do not have to be simple, but we shall assume that if k has multiplicity m, then it also has m linearly independent left eigenvectors. This is invariably observed to be the case in practice. Suppose that c of the eigenvalues of Q() are strictly inside the unit disk (each counted according to its multiplicity), while the others are on the circumference or outside. Let the numbering be such that j k j < 1 for k = 1; 2; : : :; c. The corresponding independent eigenvectors are 1, 2, : : :, c. Then any solution of equation (2.5) which can be normalised to a probability distribution is of the form v j = cx k=1 x k k j k ; j = M; M + 1; : : : ; (2.1) where x k (k = 1; 2; : : :; c ), are arbitrary (complex) constants. So far, we have obtained expressions for the vectors v M ; v M+1 ; : : :, which contain c unknown constants. Now it is time to consider equations (2.4), for j = ; 1; : : :; M. This is a set of (M + 1) N linear equations with M N unknown probabilities (the vectors v j for j = ; 1; : : :; M? 1 ), plus the c constants x k. However, only (M + 1) N? 1 of these equations are linearly independent, since the generator matrix of the Markov process is singular. On the other hand, an additional independent equation is provided by (2.6). This set of (M +1)N equations with M N + c unknowns has a unique solution when c = N. In other words, the ergodicity condition for the Markov process X is that the number of eigenvalues of Q() strictly inside the unit disk is equal to the number of operative states. In summary, the SE solution procedure consists of the following steps: (1) Compute the eigenvalues, k, and the corresponding left eigenvectors, k, of Q(). If c < N, stop; a steady-state distribution does not exist. (2) Solve the nite set of linear equations (2.4) and (2.6), with v M and v M+1 given by (2.1), to determine the constants x k and the vectors v j for j < M. (3) Use the obtained solution for the purpose of determining various moments, marginal probabilities, percentiles and other system performance measures that may be of interest. The numerical implementation of step 1 is best done by reducing the quadratic eigenvalue-eigenvector problem (2.9) to a linear one of the form yq = y, where Q is a matrix whose dimensions are twice as large as those of Q, Q 1 and Q 2 (see [1]). The latter problem is normally solved by applying

5 TIPP and the Spectral Expansion Method 5 various transformation techniques. Ecient routines for linear eigenvalueeigenvector problems are available in most numerical packages. The generalisation of the Spectral Expansion method to models with batch arrivals and/or departures is quite straightforward. The matrices A (`) j are dened for superscripts in some range?s 1 ` S 2, where S 1 and S 2 are the largest possible downward and upward jumps of the random variable J, respectively. Again, these are assumed to be independent of j above some threshold, M. The equilibrium probabilities satisfy a vector dierence equation of order S 1 + S 2. The eigenvalues in the interior of the unit disk, and the left eigenvectors, of the corresponding characteristic polynomial, provide the Spectral Expansion solution. 3. SE{TIPP The standard TIPP language was designed to model systems with a nite number of states. The description of innite state systems using TIPP involves recursion over static operators [3] which will lead to a unsuitable structure of the state space. Therefore, in order to allow such descriptions, we propose a dialect of TIPP which will be referred to as SE-TIPP and allows to specify innitely long process terms. It has the following features: { Subsystems which involve an innite number of states (such as unbounded queues) can be modeled in a very intuitive way. { The specication scheme for these subsystems leads to an economic representation of an innite Markov process which can be solved by the Spectral Expansion method. We will present a compositional approach to determine the semantics of systems which involve such innite subsystems. 3.1 Syntax Subsystems involving nite state spaces can be specied by using standard TIPP-syntax: P ::= X (a; ):P P + P P ks P P nl recx : P (3.1) X is from a set of variables V ar, a 2 Act denotes an action, its rate, and S is a set of actions to synchronise on. To guarantee the nite state property of such processes, we restrict to the class of so called rs-free 1 processes, i.e., each subterm of the form recx : P must not contain either hiding or parallel operator [3]. A process which involves an innite state space may be specied by an innite number of process denitions having the form 1 `rs' abbreviates `recursion through static operators'.

6 6 I. Mitrani et al. Q := : : : Q 1 := : : :. Q i := : : : i M: (3.2) Each process Q i may contain references to process variables Q j in order to reference the behaviour of another process Q j. Thus, this system of process terms can be considered as an equation system, recursively dening one process Q of innite length. Using such equation systems, the denition of, for example, an unbounded queue is quite easy: Q = Q := (arr; ):Q 1 Q i := (arr; ):Q i+1 + (deq; ):Q i?1 i 1 (3.3) The innite number of reachable states in this system is due to the innite length of the process term Q, and is not caused by replicating static operators, as it would be the case when modelling the queue in standard TIPP. The compact structure of the state-space of a so-dened process can only be guaranteed if all processes Q i themselves are rs-free. For simplicity, we restrict in this paper to Q i having the form P ::= (a; ):Q j P + P: (3.4) The language SE{TIPP (which will be denoted as the set L 1 of process terms) holds all processes which result from parallel composition of an arbitrary number of nite-state processes and at most one process which involves an innite state space, i.e. System := P 1 k S1 : : : k Sn?1 P n k Sn Q; (3.5) where P 1 ; : : : ; P n are nite processes, and Q is either a nite state process or a innite-state process specied by an equation system as mentioned above. 3.2 Semantic model Normally, the semantics of both nite- and innite-state processes are formally represented by labelled transition systems. However, transition systems do not impose any structure upon their state space, and thus the application of the Spectral Expansion solution method to their underlying Markov chain would represent a dicult task. To avoid this problem, we introduce the notion of generator systems. Generator systems will be used as an alternative way for describing the formal semantics of a process, essentially holding the same information as labelled transition systems, but providing additional information about the structure of the state space. Generator systems are closely related to the innitesimal generator matrices of the underlying Markov chains, thus their name. According to the division of processes in nite and innite state we come up with nite and innite generator systems.

7 TIPP and the Spectral Expansion Method 7 Finite generator systems reect the semantics of nite-state processes. A nite generator system of order N for a nite-state process P is a pair (P ; m); (3.6) where P 2? 2 ActRateLab NN is a matrix holding sets of transitions between reachable states in the transition system of process P. The bijective function m : f; : : :; N? 1g! reach(p ) associates rows and columns of this matrix with the corresponding states in the transition system. The matrix P has to hold exactly the transitions between reachable states in the transition system, i.e. 2 8i; j 2 f; : : : ; N? 1g : [P] i;j = f(a; ; w) m(i) a;;w???! m(j)g: (3.7) Since we intended to hold all information which is given in the labelled transition system, we also have to keep track of its starting state. This is accomplished by requiring m() = P: (3.8) Innite generator systems To describe the behaviour of innite-state processes, we could extend the notion of nite generator systems to hold innite matrices. However { having in mind a later analysis using the Spectral Expansion method { we propose a dierent description scheme, which is closely related to the matrices A (d) j presented in section 2. Innite generator systems (fq (d) j j 2 N ; d 2 Zg; n) (3.9)? of order N hold an innite number of matrices Q (d) j 2 2 ActRateLab NN and an injective function n : f; : : :; N? 1g N! L 1 for which img(n) reach(q) holds. This function associates each reachable state in the transition system with an ordered pair (i; j). All transitions between states (i 1 ; j 1 ) and (i 2 ; j 2 ) are kept in the matrix element [Q (j2?j1) j 1 ] i1;i 2, that is, 8i 1 ; i 2 2 f; : : :; N? 1g; 8j 2 N ; 8d 2 Z: [Q (d) j ] i1;i 2 = f(a; ; w) n(i1 ; j)???! a;;w n(i 2 ; j + d)g: (3.1) Again, to keep track of the transition system's starting state, we require n(; ) = Q: (3.11) 3.3 Construction schema We will present a compositional approach to derive the generator system semantics of a valid SE-TIPP process as given in (3.5). Thus, we describe how to get generator systems for the nite and innite components composed in 2 [A] i;j denotes the entry in row i and column j of matrix A, where counting starts with.

8 8 I. Mitrani et al. (3.5), as well as an algorithm for deriving the generator system of a composed process from the generator systems of the processes involved in the composition Component generator systems. Finite state processes The generator system (P ; m) of a nite state process P is immediately given by its transition system. The function m may be dened arbitrarily as long as m() = P and img(m) = reach(p ) hold. Given this function, the construction of the matrix P is obvious. Innite state processes If the specication of an innite state process Q is given by an equation system consisting of processes which comply to (3.4), the corresponding generator system (fq (d) j j 2 N ; d 2 Zg; n) is constructed as follows: { Since all processes Q i only contain transitions to processes Q j the condition reach(q) fq ; Q 1 ; : : :g holds. Thus we set the order of the innite generator system to 1, and dene n(; i) = Q i : (3.12) { The matrices Q i have to hold all transitions of the process Q. Since all states in the transition system of Q have the form Q i, and all transitions leaving a state Q i are dened in the equation corresponding to Q i, all transitions are included if the matrices Q i full 8i 2 f; 1; : : :g : Q i (a;;w)????! Q j =) (a; ; w) 2 [Q (j?i) i ] ; : (3.13) Parallel composition. As shown in (3.5), the complete system description emerges from a parallel composition of nite and innite subsystems. In the following, we present a compositional approach to determine the generator system of a process R = P k S Q, given the nite generator system (P ; m) of order N P of process P and Q's innite generator system (fq (d) j j 2 N ; d 2 Zg; n) of order N Q. Originally, the state space structure of R = P k S Q is three-dimensional, due to the fact that both process P and Q can act independent of each other in parallel composition. However, this three-dimensional structure can be reduced to a two dimensional one. The idea is to combine the nite number of states in reach(p ) with the states in the nite component of the state space in Q's generator system. Thus, the dimension N R of the resulting (innite) generator system (fr (d) j j 2 N ; d 2 Zg; o) is N P N Q. Figure 3.1 illustrates this idea for P = recx : (deq; 1):(wrk; ):X (thus N P = 2) and the innite queue Q modelled in (3.3) (N Q = 1), synchronised by action deq. The lower half of the state space corresponds to those states where process P is in state P, the upper half corresponds to those states where process P is in state (wrk; ):P.

9 TIPP and the Spectral Expansion Method 9 1 arr arr arr deq deq deq wrk wrk wrk arr arr arr 1 2 Fig Parallel composition of a nite and an innite generator system. ;(;u). ;(N Q?1;u) 8 ; (;u+d) ; (N Q?1;u+d) N P?1; (;u+d) N P?1; 9 (N Q?1;u+d).... N P?1;(;u). N P?1;(N Q?1;u) >: Fig Structure of the matrices R (d) u. >; Figure 3.2 shows the structure of the matrices R (d) u and the states of P and Q associated with their rows and columns (separated by comma). Depending on whether a transition of process R is due to an unsynchronised action in P, an unsynchronised action in Q, or a synchronised action which has to occur in both P and Q, three cases have to be considered in the construction of the matrices R (d) j. First, we introduce some abbreviations needed in the treatment of? these cases. Let A; B 2 2 ActRateLab NN and S Act. Then { A:S denotes the limitation of A on S, with [A: S] i;j := f(a; ; w) (a; ; w) 2 [A]i;j ^ a 2 Sg: (3.14) { AnS := A:(ActnS). { u A denotes labelprexing, with [u A] i;j := f(a; ; w) (a; ; v) 2 [A]i;j ^ w = uvg: (3.15) { A \ k B denotes parallel intersection of A and B. Parallel intersection combines rates and labels according to the SOS-rules for the parallel operator:

10 1 I. Mitrani et al. [A \ k B] i;j := f(a; ; w) (a; ; u) 2 [A] i;j ^ (a; ; v) 2 [B] i;j ^ = ^ w = (u; v)g: (3.16)? The relation between matrices in 2 ActRateLab NN is fullled if the corresponding element-wise comparisons are fullled. full N (M) denotes a matrix holding M in all its elements, and diag N (M) contains the set M on its main diagonal. The construction of the matrices R (d) u handles the three cases mentioned above 3 : 1. Unsynchronised transitions in process Q are independent of transitions in P, thus 8u 2 N ; 8d 2 Z; 8p 2 f; : : :; N P? 1g :? kr (Q (d) u ns) R (d)[p;p] u : (3.17) 2. Unsynchronised transitions in process P occur without changing the state of Q, thus they appear in all matrices Q () u : 8u 2 N : 8p 1 ; p 2 2 f; : : : ; N P? 1g :? kl diag NQ ([PnS] p1;p 2 ) R ()[p1;p2] u : (3.18) 3. Transitions with synchronising actions must occur in both processes P and Q, thus 8u 2 N : 8d 2 Z: 8p 1 ; p 2 2 f; : : :; N P g :? fullnq ([P : S] p1;p 2 ) \ k (Q (d) u : S) R (d)[p1;p2] u : (3.19) 4. Application example In this section, we will apply the presented specication and evaluation techniques to a problem originating from an ATM/B-ISDN-based communication infrastructure, which was also investigated in [5] using Stochastic Petri Nets. There, the AAL-(ATM adaption-) layers oer connectionless trac over a connection oriented communication system [7]. When packets arrive at the AAL service access points, they may suer possible delays if the connection has to be established. Once it has been established, all packets remaining in the buer can be transmitted without further connection-setup delay. When all packets are transmitted (i.e., the buer is empty), the connection can be released after a certain time. This release time heavily inuences the cost/performance ratio of the oered connectionless service: if the connection is released too quick, it has 3 For simplicity, the notation R (d)[p 1;p2] u is used for the N Q-dimensional submatrix in row p 1 and column p 2 of the matrix R (d) u. These submatrices correspond to the dotted areas in Figure 3.2.

11 TIPP and the Spectral Expansion Method 11 to be established again for packets arriving soon after the release timeout. If the release rate is too low, connections are maintained when they are not needed, thus causing unnecessary costs. 4.1 System description Arrival Process A We suppose that packets arrive in bursts, occuring with rate and ending with rate. When in burst mode, packets arrive with rate. Associating the actions on, o and arr with these events, the arrival process can be modelled as A = A := (on; ):A 1 A 1 := (o; ):A + (arr; ):A 1 : (4.1) Buer Q We assume an innite buer, whose specication is similar to the one given in (3.3). However, we model the rates passive with rate 1, so they can be determined by the other system components. Furthermore, since the behaviour of the AAL-layer depends on whether the queue is empty, the buer process indicates a non-empty queue by oering immediate transitions with action ne: Q = Q := (arr; 1):Q 1 Q i := (arr; 1):Q i+1 + (deq; 1):Q i?1 + (ne; 1):Q i i 1: (4.2) AAL-layer L The behaviour of the AAL-layer is as follows: if the buer is non-empty, connections are established with rate c (action name con). If a connection is established, packets from the buer are dequeued (action deq) and then delivered at rate (action del). Once the buer is empty, the connection is released with rate r (action rel). L = L empty := (ne; 1):L nonempty L nonempty := (con; ):L connected L connected := (deq; 1):(del; ):L connected + (rel; r):l empty : (4.3) Since packets from the buer are dequeued with an innite rate, a release action can only occur if there are no packets in the buer. Composition The complete system description is given by composing all three components: System := A k farrg Q k fne;deqg L: (4.4) 4.2 System semantics First, we demonstrate the construction scheme for R := A k farrg Q. Process A has to be associated with row of its corresponding generator system

12 12 I. Mitrani et al. matrix, and we associate process A 1 with row 1. Therefore, the matrix A is as follows 4 : ; A = : (4.5) fong fog farrg The matrices of the innite generator system for process Q are Q (?1) = [;] Q () = [;] Q (1) = [farrg] Q (?1) i = [fdeqg] Q () i = [fneg] Q (1) i = [farrg]; (4.6) where i 1. Their composition with A results in an innite generator system for process R, with matrices R (?1) ; ; = R () ; fong = R (1) ; ; = ; ; fog ; ; farrg (4.7) R (?1) fdeqg ; i = R () fneg fong ; fdeqg i = R (1) ; ; fog fneg i = ; farrg Similarly, the innite generator system for the complete system specication can be obtained by combining the generator system in (4.7) with the generator system for process L. This results in an innite generator system of order N = System evaluation Two steps have to be accomplished in order to derive the Spectral Expansion matrices A (d) j from the system's innite generator system: { Markov chains only allow nite transition rates. Thus, instantaneous transitions have to be eliminated. This is easily done by applying the corresponding TIPP-axioms to the system (see [4] for a discussion of immediate transitions). { Multiple transitions between states of the generator system have to be replaced by one transition with the sum of all rates in the Markov chain. The elimination of instantaneous transitions, and the parallel composition over a non-empty set of synchronising actions may lead to unreachable states in the Markov chain. Due to the regular structure of the Markov chain, unreachable states can be removed easily. In the present example, we reduced the system from N = 8 to N = 4, with M = 1. Two system characteristics were investigated: { The mean connection setup rate E[C]. { The mean number of packets in the buer, E[P ]. 4 For shortness, labels and rates were omitted in the following matrices.

13 TIPP and the Spectral Expansion Method conf. A conf. B.5.4 E[C] Fig The mean connection setup rate as a function of the arrival rate. All experiments were carried out with = :4, = 1: and c = 1. Two dierent congurations were chosen for the other parameters. Conguration A ( = 336; r = 1) gives preference to high transmission speeds, while con- guration B ( = 236; r = :5) favours a low connection release rate. The results for the above mentioned measures are shown in Figures 4.1 and Conclusion We presented a convenient means to model the behaviour of innite-state systems using the Stochastic Process Algebra TIPP. The approach allows the analysis of the system's underlying Markov chain with the SE solution method, thus providing an ecient solution algorithm for the presented class of systems. It has been shown that the modular design approach of Stochastic Process Algebras not only facilitates the specication of complex systems, but also simplies the application of certain solution algorithms. Concerning system specication, further work will focus on the extension of SE-TIPP to a larger class of processes by providing a completely compositional generator system semantic. Concerning system evaluation, there are several ways to improve the SE method, e.g. by allowing an arbitrary structure of the Markov chain below the limit M. This way, the problem of special treatment for unreachable states in the Markov chain can be solved.

14 14 I. Mitrani et al. 1 1 conf. A conf. B E[P ] Fig Mean number of packets in buer as a function of the arrival rate. References 1. R. Chakka and I. Mitrani (1995): Spectral Expansion Solution for a Class of Markov models: Application and Comparison with the Matrix-Geometric Solution, Performance Evaluation, N. Gotz, H. Hermanns, U. Herzog, V. Mertsiotakis, M. Rettelbach (1995). Stochastic Process Algebras { Constructive Specication Techniques Integrating Functional, Performance and Dependability Aspects, in F. Bacelli, I. Mitrani, editors, Quantitative Modelling in Parallel Systems, Springer 3. H. Hermanns, M. Rettelbach (1994): Syntax, Semantics, Equivalences and Axioms for MTIPP, in U. Herzog, M. Rettelbach, editors, Proc. of the 2nd Workshop on Process Algebra and Performance Modelling. University of Erlangen- Nurnberg, IMMD 4. H. Hermanns, M. Rettelbach, T. Wei (1995): Formal Characterisation of Immediate Actions in SPA with Nondeterministic Branching, in S. Gillmore, J. Hillston, editors, Proc. of the 3rd Workshop on Process Algebra and Performance Modelling, Springer 5. B. M. Haverkort (1995): Matrix-Geometric Solution of Innite Stochastic Petri Nets. In Proc. IEEE International Computer Performance and Dependability Symposium. Erlangen 6. I. Mitrani and D. Mitra (1991): A Spectral Expansion Method for Random Walks on Semi-Innite Strips, IMACS Symposium on Iterative Methods in Linear Algebra, Brussels 7. R. O. Onvural (1994): Asynchronous Transfer Mode: Performance Issues. Artech House 8. N.U. Prabhu and Y. Zhu (1989): Markov-Modulated Queueing Systems, QUESTA, 5,