Reversed Compound Agent Theorem Research Articles

Product-Forms in Multi-Way Synchronizations

A new algorithm is given to find product-form solutions for the joint equilibrium probabilities in a class of synchronized Markov processes.This is based on, and proved by, multiple applications of the Reversed CompoundAgent Theorem (RCAT) and can describe multi-way synchronizations (seen as chains of pairwise synchronizations) that occur in a prescribed order. The length of the sequence is unbounded but finite with probability 1. Several applications are given to illustrate the methodology, which include various modes of resets in queueing networks with negative customers. In particular, it is shown that there is a type of reset that can propagate further transitions in a chain actively. Furthermore, a number of completely new product-form models, for example, where the transitions in a chain are non-homogeneous, are given.

Product-forms in batch networks: Approximation and asymptotics

It is shown that a Markovian queue, with bulk arrivals and departures having any probability mass functions for their batch sizes, has geometrically distributed queue length at equilibrium (when this exists) provided there is an additional special bulk arrival stream, with particular rate and batch size distribution, when the server is idle. It is shown that the time-averaged input rate of the special arrivals tends to zero as the queue becomes saturated, and a heavy-traffic limit for the queue without special arrivals is derived by martingale methods. This is shown to give the same asymptotic queue length probabilities as the geometric model. The product form is then extended to tandem networks of batch queues using the reversed compound agent theorem (RCAT). In order to obtain the product form in this case, it is required that, in addition to special arrival streams, so-called ‘partial batches’ are discarded immediately from the network when there are not enough customers in the queue to fill an entire departing batch. Somewhat surprisingly it turns out that, in heavy traffic, the product-form network does not always agree with the regulated Brownian motion (RBM) diffusion limit for the standard network without special arrivals and where partial batches are not discarded, but forwarded to the next node. Indeed, we show that the two models agree in heavy traffic if and only if the skew-symmetry condition for the RBM to have a product form is satisfied. When the condition does hold, our theoretical and numerical results thus validate the use of the product-form batch networks as moderate-traffic approximations to the analogous standard queueing network model without special arrivals and where partial batches may be forwarded to the next node instead of being lost. In the case that the condition does not hold, we obtain a new product-form stationary distribution for the associated non-RBM diffusion limit.

Separable solutions for Markov processes in random environments

In this paper we address the problem of efficiently deriving the steady-state distribution for a continuous time Markov chain (CTMC) S evolving in a random environment E. The process underlying E is also a CTMC. S is called Markov modulated process. Markov modulated processes have been widely studied in literature since they are applicable when an environment influences the behaviour of a system. For instance, this is the case of a wireless link, whose quality may depend on the state of some random factors such as the intensity of the noise in the environment. In this paper we study the class of Markov modulated processes which exhibits separable, product-form stationary distribution. We show that several models that have been proposed in literature can be studied applying the Extended Reversed Compound Agent Theorem (ERCAT), and also new product-forms are derived. We also address the problem of the necessity of ERCAT for product-forms and show a meaningful example of product-form not derivable via ERCAT.

Analysis of stochastic Petri nets with signals

Product-form models facilitate the efficient analysis of large stochastic models and have been sought after for some three decades. Apart from the dominating work on queueing networks, some product-forms were found for stochastic Petri nets (SPNs) that allow fork-join constructs and for queueing networks extended to include special customers called signals, viz. G-networks. We appeal to the Reversed Compound Agent Theorem (RCAT) to prove new product-form solutions for SPNs in which there are special transitions, the firings of which act in a similar way to signals in G-networks, but which may be generated by synchronised firings (or service completions) and may affect several places simultaneously. We show that SPNs with signals are strict generalisations of G-networks with negative customers, triggers and catastrophes, and illustrate with copious examples.

Algorithmic product-form approximations of interacting stochastic models

A large variety of product-form solutions for continuous-time Markovian models can be derived by checking a set of structural properties of the underlying stochastic processes and a condition on their reversed rates. In previous work (Marin and Vigliotti (2010) [9]) we have shown how to derive a large class of product-form solutions using a different formulation of the Reversed Compound Agent Theorem (GRCAT). We continue this line of work by showing that it is possible to exploit this result to approximate the steady-state distribution of non-product-form model interactions by means of product-form ones.

Product-form approximation of queueing networks with phase-type service

The applicability of queueing network models to real-world systems is often enhanced by characterizing service times using phase-type (PH) distributions. Unfortunately, the combinatorial explosion of the underlying state space makes it difficult to evaluate efficiently the resulting models. We propose to tackle this issue by a solution algorithm inspired by the Reversed Compound Agent Theorem (RCAT) [2]. Let the QBD process for the uphill queue be specified by the forward transition matrix F , by the local transition matrix L, and by the backward transition matrix B. Also, let us indicate with αn the vector of state probability when the queue size is n. Further, let (β,T ) be the representation of the PH distribution at the uphill queue. Then a RCAT product-form exists if there exist a scalar q > 0 such that αnL = αn+1B, n ≥ 1, where L = diag(qδ1, qδ2, . . . , qδK) and δk is 1 if the kth column of −Teβ has at least one nonzero element, 0 otherwise [1]. An interesting observation arising in [1] from the above formula is that, for models with a RCAT product-form, there exist a matrix H = −F (L+ L) such that αn = αn−1H , n ≥ 2. In general, this matrix differs from the rate matrix R, even though they must share the same dominating eigenvalue η and αn = αn−1R. Observe now that αn+1 = αnR = αnH, n ≥ 1, thus αn(R −H) = 0, which implies αn(L − RB) = 0, n ≥ 1. Since the last equation must hold for all n ≥ 1, it suggests that the general solution admits the form αn+k = cn+kαn+k−1, for some sequence of coefficients ci, i > 1. Applying the Cayley-Hamilton theorem to R, it can be shown that choosing cn+k ≡ η such that αn+1 = ηαn provides always a solution that is consistent with the matrix geometric one for all n ≥ 1. In particular, if H and R share only the eigenvalue η, the relation can be proved exactly. Based on the above observations, one may then approximate the steady state of a queue satisfying RCAT’s conditions by the relation αn = ηαn−1 which, upon imposing

Methodological construction of product-form stochastic Petri nets for performance evaluation

Product-forms in Stochastic Petri nets (SPNs) are obtained by a compositional technique for the first time, by combining small SPNs with product-forms in a hierarchical manner. In this way, performance engineering methodology is enhanced by the greatly improved efficiency endowed to the steady-state solution of a much wider range of Markov models. Previous methods have relied on analysis of the whole net and so are not incremental—hence they are intractable in all but small models. We show that the product-form condition for open nets depends, in general, on the transition rates, whereas closed nets have only structural conditions for a product-form, except in rather pathological cases. Both the “building blocks” formed by the said small SPNs and their compositions are solved for their product-forms using the Reversed Compound Agent Theorem (RCAT), which, to date, has been used exclusively in the context of process-algebraic models. The resulting methodology provides a powerful, general and rigorous route to product-forms in large stochastic models and is illustrated by several detailed examples.

A unifying approach to product-forms in networks with finite capacity constraints

In queueing networks with blocking, stations wishing to transmit customers to a full queue are blocked and need to take alternative action on completing a service. In general, product-forms, i.e. separable solutions for such a network's equilibrium state probabilities, do not exist but some product-forms have been obtained over the years in special cases, using a variety of techniques. We show that the Reversed Compound Agent Theorem (RCAT) can obtain these diverse results in a uniform way by its direct application, so unifying product-forms in networks with and without blocking. New product-forms are also constructed for a type of blocking we call `skipping', where a blocked station sends its output-customers to the queue after the one causing the blocking in that customer's path. Finally, we investigate a novel congestion management scheme for networks of finite-capacity queues in which a station with a full queue transmits signals that delete customers from upstream queues in order to reduce incoming traffic.

Product-forms and functional rates

A recent result of Fourneau et al. derives conditions under which the joint state of a set of Markov chains with functional rates has a product-form solution for its equilibrium state probabilities, when they exist. The present note shows how the Reversed Compound Agent Theorem (RCAT) obtains the same result in a few lines. In fact a more general result also follows.

Process Algebraic Non-product-forms

A generalization of the Reversed Compound Agent Theorem of Markovian process algebra is derived that yields separable, but non-product-form solutions for collections of interacting processes such as arise in multi-class queueing networks with Processor Sharing servers. It is based on an analysis of the minimal cycles in the state space of a multi-agent cooperation, which can be simply identified. The extended methodology leads to what we believe are new separable solutions and, more generally, the results represent a viable practical application of the theory of Markovian process algebras in stochastic modelling.

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.

Reversed processes, product forms and a non-product form

The equilibrium joint state probabilities of interacting Markov processes are obtained in a hierarchical way, by finding the reversed process of the interaction in terms of the reversed processes of its components. From a reversed process, a product-form solution for the joint state probabilities follows directly. The method uses a Markovian process algebra formalism and generalises the recent Reversed Compound Agent Theorem (RCAT) to solve a diverse class of concurrent systems. This class includes processes with shared, exclusive resources, a customer-oriented specification of a last-come-first-served (LCFS) queue with Coxian service times and an extended PS queue with a non-product form solution. From these results, a new, very short proof of the BCMP theorem ensues. The principal advantage of the methodology is its potential for mechanisation and symbolic implementation. Indeed, many non-standard product-forms have emerged directly from the compositional approach.