BCMP Networks Research Articles

Proposal for optimizing number of servers in closed BCMP queueing network

AbstractIn this study, we use a closed BCMP queueing network model designed for multiple customer classes and servers to optimize the number of servers at each node. This optimization is achieved by setting an upper limit on the number of servers and using an objective function that combines the standard deviation of the average number of customers in the system with the server installation cost. We use a genetic algorithm with parallel computations for the optimization process. Our findings demonstrate that this approach is viable for closed BCMP network models that require extensive computational resources. The optimal server count is validated by comparing the optimization results with the maximum number of servers utilized. Node popularity is predetermined, and a gravity model is employed to generate transition probabilities, rendering the model applicable to real-world scenarios. Our optimization results indicate that both the node popularity and distance between nodes influence the server count. Furthermore, simulations were conducted to evaluate the effect of the number of servers on the optimization outcomes. Allowing variations in the node count, location, and popularity makes this study flexible and adaptable to various real-world scenarios, such as transportation systems, healthcare facilities, and commercial spaces. Moreover, by providing an efficient and scalable solution, this study serves as a cornerstone for future research in this field and offers a practical tool for facility managers aiming to minimize both congestion and operational costs.

Modeling and Simulation of Adaptive Traffic Control System for Multi-Intersection Management using Cellular Automaton and Queuing System

During last years, urban traffic has become one of the most studied research topics. This is mainly due to the enlargement of the cities and the growing number of vehicles traveling in this road network. One of the most sensitive problems is to verify if the intersections are congestion-free. Another related problem is the automatic reconfiguration of the network without building new roads to alleviate congestions. These problems require an accurate model to determine the steady state of the traffic. The present article proposes an adaptive traffic light system based on the BCMP network queuing and cellular automata. The aim of this work is to predict the best red and green time span by combining three important factors: The queue length, the evacuation time and the capacity of the destination roads. This approach can maximize the number of vehicles passing intersection and at the same time can minimize the average waiting time of vehicles as a result reducing the congestion and keep the fluency in intersections. To validate our results, we compared our model with a fixed model to explain the strengths of our proposed algorithm.

Failure detectors for crash faults in cloud

Failure detector (FD) is an inherent component in atomic broadcast and consensus protocols. Failures are broadly categorized into two types: crash and byzantine. The crash failures simply discontinue the working of a system whereas byzantine reflects the malicious behavior while ongoing communication. The problem to detect a failure becomes more challenging in a dynamic asynchronous environment like cloud computing. The paper proposes a failure detector in order to handle crash faults in cloud while addressing scalability. We introduce BCMP networks in order to compute performance parameters of the proposed algorithm, thereby, detecting failures in an accurate manner. Although, failure detection schemes have a tradeoff between efficiency and latency, the proposed algorithm achieves optimal balance between both metrics.

CONTAINER TERMINAL AS A QUEENING TRANSPORT NETWORKS

The queuing networks consist of several connected simple queuing system. In container terminal as an open network containers enter the queuing network from sea and landside, received adequately service at system and leave the network. Jackson’s network is simplest form of queuing networks, characterized by unlimited overall numbers of customers belong to the one class of customers. Customer – containers arrived' system with exponential arrival pattern (Poison or Erlang) and are served on single or multi-channel servers with exponential service time. Another case of queuing networks is network of Kelly with different classes of customers (containers) but each class of containers has fixed route (service model) in the transport system. The Baskett, Chandy, Muntz and Palacios – BCMP networks include different class of customers, with different queuing discipline and generally distributed time. Routes (service model) through the network may depend on the class of customers – containers, and the containers can change its class while passing through the system. This article describes different queuing networks, which can be used to modelling of the container terminal as a queuing transport network. The Queuing Theory may be used to determine the appropriate level of capacity required at container terminal and the staffing levels required at service facilities, over the nominal average capacity required to service expected demand. The model of the container terminal called a queuing network had better represent the real structure then a single system.

Non-product form equilibrium probabilities in a class of two-station closed reentrant queueing networks

While many single station queues possess explicit forms for their equilibrium probabilities, queueing networks are more problematic. Outside of the class of product form networks (e.g., Jackson, Kelly, and BCMP networks), one must resort to bounds, simulation, asymptotic studies or approximations. By focusing on a class of two-station closed reentrant queueing networks under the last buffer first served (LBFS) policy, we show that non-product form equilibrium probabilities can be obtained. When the number of customer classes in the network is five or fewer, explicit solutions can be obtained. Otherwise, we require the roots of a characteristic polynomial and a matrix inversion that depend only on the network topology. The approach relies on two key points. First, under LBFS, the state space can be reduced to four dimensions independent of the number of buffers in the system. Second, there is a sense of spatial causality in the global balance equations that can then be exploited. To our knowledge, these two-station closed reentrant queueing networks under LBFS represent the first class of queueing networks for which explicit non-product form equilibrium probabilities can be constructed (for five customer classes or less), the generic form of the equilibrium probabilities can be deduced and matrix analytic approaches can be applied. As discussed via example, there may be other networks for which related observations can be exploited.

G-Networks: Development of the Theory of Multiplicative Networks

This is a review on G-networks, which are the generalization of the Jackson and BCMP networks, for which the multi-dimensional stationary distribution of the network state probabilities is also represented in product form. The G-networks primarily differ from the Jackson and BCMP networks in that they additionally contain a flow of the so-called negative customers and/or triggers. Negative customers and triggers are not served. When a negative customer arrives at a network node, one or a batch of positive (ordinary) customers is killed (annihilated, displaced), whereas a trigger displaces a positive customer from the node to some other node. For applied mathematicians, G-networks are of great interest for extending the multiplicative theory of queueing networks and for practical specialists in modeling computing systems and networks and biophysical neural networks for solving pattern recognition and other problems.

Insensitivity and product-form decomposability of reallocatable GSMP

A stochastic process, called reallocatable GSMP (RGSMP for short), is introduced in order to study insensitivity of its stationary distribution. RGSMP extends GSMP with interruptions, and is applicable to a wide range of queues, from the standard models such as BCMP and Kelly's network queues to new ones such as their modifications with interruptions and Serfozo's (1989) non-product form network queues, and can be used to study their insensitivity in a unified way. We prove that RGSMP supplemented by the remaining lifetimes is product-form decomposable, i.e. its stationary distribution splits into independent components if and only if a version of the local balance equations hold, which implies insensitivity of the RGSMP scheme in a certain extended sense. Various examples of insensitive queues are given, which include new results. Our proofs are based on the characterization of a stationary distribution for SCJP (self-clocking jump process) of Miyazawa (1991).

Insensitivity and product-form decomposability of reallocatable GSMP

A stochastic process, called reallocatable GSMP (RGSMP for short), is introduced in order to study insensitivity of its stationary distribution. RGSMP extends GSMP with interruptions, and is applicable to a wide range of queues, from the standard models such as BCMP and Kelly's network queues to new ones such as their modifications with interruptions and Serfozo's (1989) non-product form network queues, and can be used to study their insensitivity in a unified way. We prove that RGSMP supplemented by the remaining lifetimes is product-form decomposable, i.e. its stationary distribution splits into independent components if and only if a version of the local balance equations hold, which implies insensitivity of the RGSMP scheme in a certain extended sense. Various examples of insensitive queues are given, which include new results. Our proofs are based on the characterization of a stationary distribution for SCJP (self-clocking jump process) of Miyazawa (1991).

Queueing networks with dependent nodes and concurrent movements

Classical queueing network processes are useful for modeling the movement of discrete units in a network in which the nodes operate independently, the routing of units is independent of the congestion, only one unit moves at a time and its equilibrium distribution is a well-understood product form. Actual networks, however, typically have dependent nodes and concurrent movement of units. Imagine the dependencies associated with the network movements of telephone calls, manufacturing material, computer data packets, messages in a parallel-processing simulation, etc. A second generation of queueing network processes is beginning to evolve for modeling such “intelligent” networks with dependent nodes and concurrent movements. This paper describes the following fundamental processes that have been developed in this regard: • A basic queueing network process for dependent nodes and single-unit movements. Examples include the classical Jackson, BCMP, Kelly and Kelly-Whittle networks and networks with interacting subpopulations. • Reversible queueing network processes for dependent nodes and concurrent movements. An example is a multivariate, compound birth-death process. • Miscellaneous partially balanced queueing networks. Examples include extensions of the basic network processes and weakly coupled and quasi-reversible networks.

Operational analysis of stochastic closed queueing networks

The purpose of this article is to provide alternative derivations and explanations of some classical results on stochastic closed queueing networks with a single class of customers, using operational analysis. The results we deal with are mainly the product-form solution, the Sevcik-Mitrani theorem, the aggregation property, and Marie's approximate method. We first extend operational analysis in order to analyze the asymptotic behaviour of stochastic closed queueing networks. We then show that any realization of a closed single-class BCMP network satisfies all operational assumptions with probability one. For queueing networks with exponential service times, we provide a simple proof of the Sevcik-Mitrani theorem. As opposed to the classical derivation of the aggregation property which is based on some algebraic manipulations, we provide a direct proof of this property by showing that the aggregate network satisfies the operational assumptions yielding the product-form solution. Finally, we provide an alternative explanation of Marie's approximate method for general closed queueing networks.

Sensitivity results in open, closed and mixed product form queueing networks

General formulas are proposed to quantify the effects of changing the model parameters in the so-called BCMP network [F. Baskett et al., J. ACM 22 (2) (April 1975) 248–260]. These formulas relate the derivative of the expectation of any function of both the state and the paramaters of the network with respect to any model parameter (i.e., arrival rate, mean service demand, service rate, visit ratio, traffic intensity) to known functions of the state variables. Applications of our results to sensitivity analysis and optimization problems are given.

A New queueing network model for the simulation of flexible manufacturing systems optimization method

In the simulation of FMS(Flexible Manufacturing System), most of current approximate methods are far from satisfactory. It is because that they can not give us any reliable information about the error caused by them, or give rise to a high cost of computation burden. And even the correctness or convergence of them can not be guaranteed. To tackle the problem, this paper introduce the BCMP network (See,Baskett F, Chandy, K.M., Munttz R.R., Palacios F.G.; Open,Closed And Mixed Networks of Queues With Different Classes of Customers.JACM, 1975;22(2):248-260) in to a closed system which has only one central server, and sets up new multivariable queueing network model for the syetem. This method can not only give a measure of the error by approximation, but also minimize the error.In addition, the method is a recursive algorithm, so it requires little amount of computation.

An approximate product form for a class of degradable queueing networks

An approximate product form for a class of degradable queueing networks (DQNs) is presented. DQNs emerge from a synthesis of the well-known separable queueing networks (BCMP networks) and the classical machine-repair-man model. Under the assumption that failure and repair events which change the level of performance are much less frequent compared with the service completions of tasks moving through the network, the presented product from approximates the exact probability distribution very closely. Because DQNs are nearcompletely decomposable, the error of approximation depends on the degree of coupling between the different modes of degraded performance. In accordance with the theory of nearcomplete decomposability, a numerical example shows that the approximation error is proportional to the degree of coupling.

Modeling a Slotted Ring Local Area Network

Models for local area networks of the slotted ring style of architecture are developed and evaluated. The hardware protocol is modeled using a BCMP network. The Basic Block protocol of the Cambridge ring is modeled using an approximate solution method of the fixed-point type. A limited comparison between the Cambridge Ring and another ring architecture—the token ring—is carried out.

A BCMP extension to multiserver stations with concurrent classes of customers

We consider a multiclass service station with B identical exponential servers, with constant service rate μ. At a station, the classes of customers are sorted into M concurrent groups ; the discipline of service is on a first come first served basis, but two customers of the same group cannot be served simultaneously. We show that product form is maintained when such stations are inserted in BCMP networks, and give closed form expressions for the steady-state probabilities.

Queueing systems with resource sharing

Analytical methods for evaluating the performance of computer systems primarily use queueing network systems. However, those systems must verify independence assumptions that are essential to provide product-form queueing networks. We can't, therefore, directly represent some classical behaviors in computer systems, especially since the emergence of distributed systems (e.g., computer networks, multiprocessor architecture). Simultaneous resource possessions belong to those behaviors and can have a significant effect on system performances. In this paper, we study systems in which there is a common resource. This shared resource is modeled by an allocation queue with a limited number of servers. A customer obtains and holds one of the allocation servers for a first service time, then requests services inside a BCMP network while still keeping the server of the allocation queue busy. In a first part we find the value of the stability condition of those networks, then we introduce an approximate technique to evaluate those systems. Several practical examples are considered, and simulation is used to validate the analytical model and to test its robustness. The approximate results are compared with simulation and found to be very accurate.

Mean value analysis of mixed, multiple class BCMP networks with load dependent service stations

In this paper we describe a mean value analysis algorithm to solve product form queueing network models of the Baskett, Chandy, Muntz, and Palacios type (BCMP) [5]. These networks can accommodate multiple job classes and load dependent service stations. The networks can be closed (i.e., jobs neither enter nor leave the network), open (i.e., jobs can enter and leave the network), or mixed (i.e., closed with respect to some job classes and open with respect to others). Two major aspects of this paper are: 1. (1)to present a unified approach to the solution of this class of BCMP queueing networks [5], and 2. (2)to thereby extend and consolidate the results of the work of several authors (Balbo, Bruell and Schwetman (1977), Bruell and Balbo (1980), Chandy and Sauer (1980), Krzesinki, Teunissen and Kritzinger (1981), Resier and Kobayashi (1975), Reiser and Lavenberg (1980), Reiser (1981), Sauer (1983), Schwetman (1980), Tucci and MacNair (1982) and Zahorjan and Wong (1981)). The paper concludes by collecting together all the results into an explicit algorithm, MVALDMX (for mean value analysis load dependent mixed) that is a generalization of the original MVA algorithm of Reiser and Lavenberg (1978, 1980).