Peter G. Harrison

Parallel Programming Using Skeleton Functions

Programming parallel machines is notoriously difficult. Factors contributing to this difficulty include the complexity of concurrency, the effect of resource allocation on performance and the current diversity of parallel machine models. The net result is that effective portability, which depends crucially on the predictability of performance, has been lost. Functional programming languages have been put forward as solutions to these problems, because of the availability of implicit parallelism. However, performance will be generally poor unless the issue of resource allocation is addressed explicitly, diminishing the advantage of using a functional language in the first place.

Turning back time in Markovian process algebra

Product-form solutions in Markovian process algebra (MPA) are constructed using properties of reversed processes. The compositionality of MPAs is directly exploited, allowing a large class of hierarchically constructed systems to be solved for their state probabilities at equilibrium. The paper contains new results on both reversed stationary Markov processes as well as MPA itself and includes a mechanisable proof in MPA notation of Jacksons theorem for product-form queueing networks. Several examples are used to illustrate the approach.

The M / G /1 queue with negative customers

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer- the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

Compositional reversed Markov processes, with applications to G-networks

Stochastic networks defined by a collection of cooperating agents are solved for their equilibrium state probability distribution by a new compositional method. The agents are processes formalised in a Markovian Process Algebra, which enables the reversed stationary Markov process of a cooperation to be determined symbolically under appropriate conditions. From the reversed process, a separable (compositional) solution follows immediately for the equilibrium state probabilities. The well-known solutions for networks of queues (Jacksons theorem) and G-networks (with both positive and negative customers) can be obtained simply by this method. Here, the reversed processes, and hence product-form solutions, are derived for more general cooperations, focussing on G-networks with chains of triggers and generalised resets, which have some quite distinct properties from those proposed recently. The methodologys principal advantage is its potential for mechanisation and symbolic implementation; many equilibrium solutions, both new and derived elsewhere by customised methods, have emerged directly from the compositional approach. As further examples, we consider a known type of fork-join network and a queueing network with batch arrivals.

Passage time distributions in large Markov chains

Probability distributions of response times are important in the design and analysis of transaction processing systems and computer-communication systems. We present a general technique for deriving such distributions from high-level modelling formalisms whose state spaces can be mapped onto finite Markov chains. We use a load-balanced, distributed implementation to find the Laplace transform of the first passage time density and its derivatives at arbitrary values of the transform parameter s. Setting s = 0 yields moments while the full passage time distribution is obtained using a novel distributed Laplace transform inverter based on the Laguerre method. We validate our method against a variety of simple densities, cycle time densities in certain overtake-free (tree-like) queueing networks and a simulated Petri net model. Our implementation is thereby rigorously validated and has already been applied to substantial Markov chains with over 1 million states. Corresponding theoretical results for semi-Markov chains are also presented.

SOJOURN TIMES IN SINGLE-SERVER QUEUES WITH NEGATIVE CUSTOMERS

We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.

SPADES - a process algebra for discrete event simulation

We present a process algebra, SPADES, based on Milner’s CCS, which may be used to describe discrete event simulations with parallelism. It is able to describe the passing of time and probabilistic choice, either discrete, between a countable number of processes, or continuous, to choose a random amount of time to wait. Its operational semantics is presented as a labelled transition system and we discuss equivalences over this operational semantics that imply axioms that can be used to compare and transform processes formally. We discuss notions of equivalence over simulations and the meaning of non-determinism in the context of the specification of a simulation. The algebra is applied to describe quantitatively a range of communicating systems.

Separable equilibrium state probabilities via time reversal in Markovian process algebra

The reversed compound agent theorem (RCAT) is a compositional result that uses Markovian process algebra (MPA) to derive the reversed process of certain interactions between two continuous time Markov chains at equilibrium. From this reversed process, together with the given, forward process, the joint state probabilities can be expressed as a product-form, although no general algorithm has previously been given. This paper first generalises RCAT to multiple (more than two) cooperating agents, which removes the need for multiple applications and inductive proofs in cooperations of an arbitrary number of processes. A new result shows a simple stochastic equivalence between cooperating, synchronised processes and corresponding parallel, asynchronous processes. This greatly simplifies the proof of the new, multi-agent theorem, which includes a statement of the desired product-form solution itself as a product of given state probabilities in the parallel components. The reversed process and product-form thus derived rely on a solution to certain rate equations and it is shown, for the first time, that a unique solution exists under mild conditions--certainly for queueing networks and G-networks.

A Markov modulated multi-server queue with negative customers---the MM CPP/GE/c/L G-queue

Abstract. We obtain the queue length probability distribution at equilibrium for a multi-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a continuous time Markov phase process. In each phase, the arrivals are a superposition of a positive and a negative arrival stream, each of which is a compound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Such a queueing system is well suited to B-ISDN/ATM networks since it can account for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimensional (queue length by phase) Markov process that describes the dynamics of the system. Several variants of the system are considered, applicable to different modelling situations, such as server breakdowns, cell losses and load balancing. We also consider the departure process and derive its batch size distribution and the Laplace transform of the interdeparture time probability density function. From this, a recurrence formula is obtained for its moments. The analysis therefore provides the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes.

Exploiting Quasi-reversible Structures in Markovian Process Algebra Models

Efficient product form solution is one of the major attractions of queueing networks for performance modelling purposes. These models rely on a form of interaction between nodes in a network which allows them to be solved in isolation, since they behave as if independent up to normalisation. Markovian process algebras (MPA) extend classical process algebras with information about the duration of actions but retain their compositional structure: a system is modelled as an interaction of components. The advantages of this compositional structure for model construction and model simplification have already been demonstrated. In this paper we exploit results from queueing networks to identify a restricted form of interaction between suitable MPA components which leads to a product form solution. Each component of the model may be solved separately and the compositional structure of an MPA consequently facilitates efficient solution for successively more complex models. This work uses the notion of quasi-reversibility in a Markov process setting to define the type of interaction between MPA components. This leads to a substantial class of MPA definitions that have product-form solutions which is more general than the usual queueing network-based class of Markov processes.