Product-form Approximation Research Articles

G-network models to support planning for disaster relief distribution

One of the key activities during disaster response is distributing relief items to victims. This is a challenging task due to dynamically changing victim needs and disaster aftermath conditions. We model the distribution operations where items like tarpaulins and blankets are distributed by volunteers, to victims at temporary distribution areas called relief centers (RC). We investigate the impact victim movements have on the distribution performance. We model each RC as a queue, and the distribution operation as a generalised queuing network (G-network). We investigate product form solutions for the proposed G-network model, and prove a new product form result for G-networks with signals and batch transfer under certain conditions. We leverage this result to develop product form approximations that apply across a broad range of settings. We apply the G-network model to a case study using the Nepal earthquake relief distribution data, and quantify the impact of victim movement on network performance.

Towards a global model of spin-orbit coupling in the halocarbenes.

We report a global analysis of spin-orbit coupling in the mono-halocarbenes, CH(D)X, where X = Cl, Br, and I. These are model systems for examining carbene singlet-triplet energy gaps and spin-orbit coupling. Over the past decade, rich data sets collected using single vibronic level emission spectroscopy and stimulated emission pumping spectroscopy have yielded much information on the ground vibrational level structure and clearly demonstrated the presence of perturbations involving the low-lying triplet state. To model these interactions globally, we compare two approaches. First, we employ a diabatic treatment of the spin-orbit coupling, where the coupling matrix elements are written in terms of a purely electronic spin-orbit matrix element which is independent of nuclear coordinates, and an integral representing the overlap of the singlet and triplet vibrational wavefunctions. In this way, the structures, harmonic frequencies, and normal mode displacements from ab initio calculations were used to calculate the vibrational overlaps of the singlet and triplet state levels, including the full effects of Duschinsky mixing. These calculations have allowed many new assignments to be made, particularly for CHI, and provided spin-orbit coupling parameters and values for the singlet-triplet gaps. In a second approach, we have computed and fit full geometry dependent spin-orbit coupling surfaces and used them to compute matrix elements without the product form approximation. Those matrix elements were used in similar fits varying the anharmonic constants and singlet-triplet gap to reproduce the experimental levels. The derived spin-orbit parameters for carbenes CHX (X = Cl, Br, and I) show an excellent linear correlation with the atomic spin-orbit constant of the corresponding halogen, indicating that the spin-orbit coupling in the carbenes is consistently around 14% of the atomic value.

Explicit solutions for queues with Hypo- or Hyper-exponential service time distribution and application to product-form approximations

Queueing systems with Poisson arrival processes and Hypo- or Hyper-exponential service time distribution have been widely studied in the literature. Their steady-state analysis relies on the observation that the infinitesimal generator matrix has a block-regular structure and, hence, the matrix analytic method may be applied. Let πnk be the steady-state probability of observing the kth phase of service and n customers in the queue, with 1≤k≤K, and K the number of phases, and let πn=(πn1,…,πnK). Then, it is well-known that there exists a rate matrix R such that πn+1=πnR. In this paper, we give a symbolic expression for such a matrix R for both cases of Hypo- and Hyper-exponential queueing systems. Then, we exploit this result in order to address the problem of approximating a M/HypoK/1 queue by a product-form model. We show that the knowledge of the symbolic expression of R allows us to specify the approximations for more general models than those that have been previously considered in the literature and with higher accuracy.

Approximation analysis of multi-class closed queueing maintenance networks with a parts inventory system and two-phase Coxian time distributions

We consider a maintenance network where a set of bases is supported by a replacement parts inventory system and a centrally located repair depot. The ordering policy for the parts is the (S, Q) inventory policy. We extended the previous results to the network, where processing times at each node follow a two-phase Coxian distribution. The proposed network was modeled as a multi-class closed queueing network with a synchronization station. To make the analysis of the network computationally tractable, we developed a two-phase approximation method. In the first phase of the method, the proposed network was analyzed with the previous algorithm based on a product-form approximation. In the second phase, a sub-network was again analyzed with the procedure of a product-form approximation method such that the state space of the sub-network was reduced. In the analysis of a sub-network, a recursive method was also used to solve balance equations by exploiting the special structure of the Markov chain. The new algorithm provided a good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.

Product-form characterization for a two-dimensional reflecting random walk

We consider a two-dimensional skip-free reflecting random walk on the non-negative integers, which is referred to as a 2-d reflecting random walk. We give necessary and sufficient conditions for the stationary distribution to have a product-form. We also derive simpler sufficient conditions for the product-form for a restricted class of 2-d reflecting random walks. We apply these results and obtain a product-form approximation of the stationary distribution through a suitable modification of the parameters of the random walk.

Poisson 수요과정을 갖는 두 품목 콘윕시스템의 성능평가

In this study we consider a flow line CONWIP system in which two types of product are produced. The processing times of each product type at each station follow an independent exponential distribution and the demands for the finished products of each type arrive according to a Poisson process. The demands that are not satisfied instantaneously are either backordered or lost according to the number of unsatisfied demands that exist at their arrival instants. For this system we develop an approximation method to obtain the performance measures such as steady state probabilities of the number of parts of each product type at each station, mean waiting times of backordered demands and the proportion of backordered demands. For the analysis of the proposed CONWIP system, we model the CONWIP system as a two class closed queueing network with a synchronization station and analyze the closed queueing network using a product-form approximation method for multiple classes developed by Baynat and Dallery. In the approximation method, each subsystem is analyzed using a matrix geometric method. Comparisons with simulation show that the approximation method provides fairly good results for all performance measures.

Performance evaluation of a multi-product CONWIP assembly system with correlated external demands

Abstract We study a multi-product assembly production system in which individual components are made to meet various order types. We assume that orders arrive according to a Poisson process, but there is a fixed probability that an order requests a particular kit of different components. Under the CONWIP control mechanism, each component is produced by a flow line with several stations. To analyze this system, we develop an approximation algorithm based on a decomposition method. In the algorithm, a product-form approximation technique as well as a matrix-geometric method is used. Numerical results show that the accuracy of the approximation method is acceptable.

A simulation model with synchronization manufacturing stations

PurposeThe aim of this paper is to develop an efficient analytical procedure to evaluate performance of the most general pull production systems particularly when multiple‐part‐types are involved. The authors consider a kanban controlled production system that can be modelled as a closed queuing network with different product classes. The production line is decomposed into stages which consist of one or several stations and an output buffer. Each stage is associated with a given number of kanbans. The main idea of this analytical algorithm is to analyze each subnetwork individually using a product form approximation technique. The iterative procedure is used to find the unknown parameters.Design/methodology/approachThe authors design a multiclass queuing network that can be used to represent kanban controlled production systems. To solve this model, three procedures are used: decompose the original network into M subnetworks, convergence of unknown parameters in each subnetwork, and convergence of unknown parameters in the original network. The authors now describe these procedures separately.FindingsThe main contribution of this paper is the formulation of the problem of kanban controlled production systems with several part‐types. The methodology is based on approximate formula with decomposition and is applicable to more general manufacturing environments. The authors' method can be applied to both limited and unlimited demands. The analytical algorithm designed in this work has demonstrated excellent performance in analyzing kanban controlled production systems.Originality/valueThe methodology of this algorithm is based on approximate formula and is applicable to more general manufacturing environments.

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

Product Form Approximation of Transient Probabilities in Stochastic Reaction Networks

Most Markov chains that describe networks of stochastic reactions have a huge state space. This makes exact analysis infeasible and hence the only viable approach, apart from simulation, is approximation. In this paper we derive a product form approximation for the transient probabilities of such Markov chains. The approximation can be interpreted as a set of interacting time inhomogeneous Markov chains with one chain for every reactant of the system. Consequently, the computational complexity grows only linearly in the number of reactants and the approximation can be carried out for Markov chains with huge state spaces. Several numerical examples are presented to illustrate the approach.

A multi-class closed queueing maintenance network model with a parts inventory system

The system we address is a maintenance network of repairable items where a set of bases is supported by a centrally located repair depot and a consumable replacement parts inventory system. If an item fails, a replacement part must be obtained at the parts inventory system before the failed item enters the repair depot. The ordering policy for the parts is the (S,Q) inventory policy. An approximation method for this system is developed to obtain performance measures such as steady-state probabilities of the number of items at each site and the expected backorders at the parts inventory system. The proposed system is modelled as a multi-class closed queueing network with a synchronization station and analyzed using a product-form approximation method. Particularly, the product-form approximation method is adapted so that the computational effort on estimating the parameters of the equivalent multi-class network is minimized. In analyzing a sub-network, a recursive method is used to solve balance equations by exploiting the special structure of the Markov chain. Numerical tests show that the approximation method provides fairly good estimation of the performance measures of interests.

A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms

A generic maximum entropy (ME) product-form approximation is proposed for arbitrary single class open first-come-first-served (FCFS) queueing network models with blocking (QNMs-B), subject to bursty GE-type interarrival and service times and the mixed blocking mechanisms (BMs) of Blocking-After-Service (BAS), Blocking-Before-Service (BBS) and Repetitive-Service (RS) Blocking with Random (RS-RD) and Fixed (RS-FD) destinations. A new GE-type analytic framework is devised, based on the ME analysis of a virtual multiple class GE/GE/1/N+U queueing system with finite capacity, $N (N>1)$ augmented by $U (U\geq1)$ auxiliary-waiting lines, to determine the first two moments of BAS- and BBS-dependent effective service times towards a node-by-node decomposition of the entire network. In this context, a unified ME algorithm is devised for the approximate analysis of arbitrary open FCFS QNMs-B with a mixture of the BMs of BAS, BBS, RS-RD and RS-FD. Typical numerical tests are carried out to assess the credibility of the unified ME algorithm against discrete event simulation and also establish GE-type experimental performance bounds. A critique on the feasibility of ME formalism for QNMs-B and suggested extensions are included.

Product form approximations for communicating Markov processes

Product form solutions have been found for several classes of stochastic models including some networks of stochastic automata or communicating Markov chains. In this paper a theory of approximate and higher order product forms is presented. The idea is to define an approximate product form solution as a Kronecker product of vectors that minimizes the Euclidean norm of the residual vector for arbitrary networks of communicating Markov processes. If the residual becomes zero, the product form becomes exact. By adopting ideas from numerical analysis to approximate a matrix by a sum of Kronecker products of small matrices, higher order product forms that result in better approximations are defined. This paper presents the general theory of product form approximations for communicating Markov processes and it introduces first algorithms to compute product form solutions. By means of some examples it is shown that the approach allows one to compute approximations with increasing accuracy by increasing the order of the product form.

Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are (i) classical circuit switch telephone networks (loss networks) and (ii) present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to upper bounds for loss probabilities and analytic error bounds for the accuracy of the approximation for various performance measures. The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications: pure loss networks as under (i) GSM networks with fixed channel allocation as under (ii). The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning.

Solving Semi-Open Queuing Networks

A semi-open queuing network (SOQN) is a special type of a queuing network consisting of two parts: an inner network with a population constraint and an external queue to accommodate jobs whose entrance is delayed. We first study an SOQN with a single class of jobs in tandem configuration and then extend our study to multiclass configurations. Multiclass SOQNs fall into two categories: general pallet and dedicated pallet SOQNs. For the general pallet case, we aggregate all classes and solve the resulting single-class SOQN. For the dedicated pallet case, we construct a method based on an existing product-form approximation method for general, multiclass closed networks. Our approximation method combines the matrix-geometric method with the decomposition-aggregation approach. Numerical results show that our approximations have desirable accuracy and efficiency.

A FINITE POPULATION MODEL AND PRODUCT-FORM APPROXIMATION FOR CELLULAR MOBILE SYSTEMS WITH TRAVELING USERS(<Special Issue>the 50th Anniversary of the Operations Research Society of Japan)

We propose a new finite population model for cellular mobile systems with traveling users. In this model, mobile users arrive according to a Poisson process from outside the system, independently travel in the system, and leave the system in due time. A mobile user is either in call (active) or out of call (inactive). We find that if two minor modifications are made to the model, the joint distribution of the number of calls in progress in each cell has the product form. Making the two modifications is referred to as product-form approximation in this paper. Under the product-form approximation, the probability that channels are fully occupied in a cell is given by the Erlang-loss formula. We evaluate the accuracy of the product-form approximation through several simulation experiments, and find that the Erlang-loss formula remains applicable to the performance evaluation and channel provisioning of cellular mobile systems.

Product form approximations for highly linear loss networks with trunk reservation

Admission controls, such as trunk reservation, are often used in loss networks to optimise their performance. Since the numerical evaluation of performance measures is complex, much attention has been given to finding approximation methods. The Erlang Fixed-Point (EFP) approximation, which is based on an independent blocking assumption, has been used for networks both with and without controls. Several more elaborate approximation methods which account for dependencies in blocking behaviour have been developed for the uncontrolled setting. This paper is an exploratory investigation of extensions and synthesis of these methods to systems with controls, in particular, trunk reservation. In order to isolate the dependency factor, we restrict our attention to a highly linear network. We will compare the performance of the resulting approximations against the benchmark of the EFP approximation extended to the trunk reservation setting. By doing this, we seek to gain insight into the critical factors in constructing an effective approximation.

Analysis of GSM/GPRS Cell with Multiple Data Service Classes

This paper is concerned with the performance modelling and evaluation of a General Packet Radio Service (GPRS) for a wireless Global System for Mobile telecommunication (GSM) cell involving both voice and multiple class data services under a complete partitioning scheme (CPS). The GPRS is modelled as a system of access and transfer queues in tandem having an external multiple class compound Poisson bursty traffic with geometrically distributed batches under first-come-first-served (FCFS) and generalised processor-sharing (PS) scheduling disciplines, respectively. The proposed queueing model is valid for both uplink and downlink connections and provides multiple class data services with different arrival rates, interarrival-time squared coefficients of variation (SCVs), file (burst) sizes and different PS discrimination service levels. Based on the principle of maximum entropy (ME), a product form approximation is characterised for the two queues in tandem, subject to queueing theoretic mean value constraints. The ME solution implies a decomposition of the system into individual single server building block queues, each of which can be analysed in isolation. Consequently, ME closed form expressions and simulation results for the aggregate state probability together with the marginal mean queue length and blocking probability are divised to assess the credibility of the ME solution and also to study the effect of external GPRS bursty traffic upon the performance of the wireless cell.

Performance modelling of GPRS with bursty multiclass traffic

An analytic framework is devised, based on the principle of maximum entropy (ME), for the performance modelling and evaluation of a wireless GSM/GPRS cell supporting bursty multiple class traffic of voice calls and data packets under complete partitioning (CPS), partial sharing (PSS) and aggregate sharing (ASS) traffic handling schemes. Three distinct open queueing network models (QNMS) under CPS, PSS and ASS, respectively, are described, subject to external compound Poisson traffic processes and generalised exponential (GE) transmission times under a repetitive service blocking mechanism and a complete buffer sharing management rule. Each QNM generally consists of three building block stations, namely a loss system with GSM/GPRS traffic and a system of access and transfer finite capacity queues in tandem dealing with GPRS traffic under head-of-line and discriminatory processor sharing scheduling disciplines, respectively. The analytic methodology is illustrated by focusing on the performance study of the GE-type tandem queueing system for GPRS under a CPS. An ME product-form approximation is characterised leading into a decomposition of the tandem system into individual queues and closed-form ME expressions for state and blocking probabilities are presented. Typical numerical examples are included to validate the ME solutions against simulation and study the effect of external GPRS bursty traffic upon the performance of the cell. Moreover, an overview of recent extensions of the work towards the analysis of a GE-type multiple server finite capacity queue with preemptive resume priorities and its implications towards the performance modelling and evaluation of GSM/GPRS cells with PSS and ASS are included.