Analysis of K-Capacity Queueing System with Two-Heterogeneous Server

This article provides an analytic solution for performance measures of finite channel capacity m queuing system M/M/m/K/N. Queuing system with a finite number of call sources size N and restricted storage K is investigated. Markov chains are used for the performance measures evaluation. The solution to our queuing system with losses, which is the birth-death system, was obtained by applying the finite Markov chains. There is used system with primitive call flow model. Interarrival time and service time represented using exponential distribution. Packets arriving to find K+m already in the system are lost and return immediately to the arriving state as if they had just completed service. The most important performance measures of queuing system are defined in an analytic form, such as delay probability P(W>0), mean value of waiting time mean value of queue length E[Nq], call loss probability B, service channel utilization ym, and etc. Taken queuing models for M/M/m/K/N systems are very elegant in analysis and provide good closed results. Some of these models were extensively used in designing telephone networks, but however they are very useful in modeling of packets or messages services in data networks. We describe the Markov birth-death model used for calculating characteristics of asymmetric queuing system M/M/2/K/N, when the fastest free channel is selected. So we examine the system linked to two channels with different packets transmission rates μ1 > μ2. Some calculation results are presented in diagram form. Ill. 15, bibl. 7 (in English; summaries in English, Russian and Lithuanian).

Many mathematical models for the disease transmission dynamics of Ebola have been developed and studied, particularly during and after the 2014 outbreak in West Africa. Most of these models are systems of ordinary differential equations (ODEs). One of the common assumptions made in these ODE models is that the duration of disease stages, such as latent and infectious periods, follows an exponential distribution. Gamma distributions have also been used in some of these models. It has been demonstrated that, when the models are used to evaluate disease control strategies such as quarantine or isolation, the models with exponential and Gamma distribution assumptions may generate contradictory results (Feng et al. in Bull Math Biol 69(5):1511-1536, 2007). Several Ebola models are considered in this paper with various stage distributions, including exponential, Gamma and arbitrary distributions. These models are used to evaluate control strategies such as isolation (or hospitalization) and timely burial and to identify potential discrepancies between the results from models with exponential and Gamma distributions.

In this paper, we study a number of closely related paradoxes of queuing theory, each of which is based on the intuitive notion that the level of congestion in a queuing system should be directly related to the stochastic variability of the arrival process and the service times. In contrast to such an expectation, it has previously been shown that, in all Hk/G/1 queues, PW (the steady-state probability that a customer has to wait for service) decreases when the service-time becomes more variable. An analagous result has also been proved for ploss (the steady-state probability that a customer is lost) in all Hk/G/1 loss systems. Such theoretical results can be seen, in this paper, to be part of a much broader scheme of paradoxical behaviour which covers a wide range of queuing systems. The main aim of this paper is to provide a unifying explanation for these kinds of behaviour. Using an analysis based on a simple, approximate model, we show that, for an arbitrary set of nGI/Gk/1 loss systems (k=1,..., n), if the interarrival-time distribution is fixed and ‘does not differ too greatly’ from the exponential distribution, and if the n systems are ordered in terms of their ploss values, then the order that results whenever cA<1 is the exact reverse of the order that results whenever cA>1, where cA is the coefficient of variation of the interarrival time. An important part of the analysis is the insensitivity of the ploss value in an M/G/1 loss system to the choice of service-time distribution, for a given traffic intensity. The analysis is easily generalised to other queuing systems for which similar insensitivity results hold. Numerical results are presented for paradoxical behaviour of the following quantities in the steady state: ploss in the GI/G/1 loss system; PW and Wq (the expected queuing time of customers) in the GI/G/1 queue; and pK (the probability that all K machines are in the failed state) in the GI/G/r machine interference model. Two of these examples of paradoxical behaviour have not previously been reported in the literature. Additional cases are also discussed.

The simple model NMEAR (1) is described for a stationary dependent sequence of random variables which have a mixed exponential marginal distribution; the model is a first-order stochastic difference equation with random coefficients and is first-order Markovian. It should be broadly applicable for stochastic modelling in operations analysis. In particular, it provides a model for simulating interarrival times in queuing systems when these random variables are overdispersed relative to an exponential random variable, and moreover are positively correlated. The model also has capability to model a variable which may be zero, but which otherwise is exponentially distributed. Such variables are found as waiting times in queuing models. Because of the (random) linearity of the process, it is easily extended to the modelling of cross-coupled sequences of interarrival and service times. The model can also be extended quite simply to a mixed exponential process with mixed pth order autoregressive and qth order moving average correlation structure, NMEAR (p, q), so that non-Markovian dependence can be handled.

This paper is concerned on an M/M/1 queue with correlated interarrival and service times. In particular, we assume that the interarrival time and service time of a customer have a bivariate exponential distribution. By utilizing a Markov modulated fluid flow (MMFF) process associated with the age process of the customer in service, we obtain a number of queueing quantities in closed form. Using the solutions, the impact of the correlation coefficient of the interarrival and service times on a variety of queueing quantities is explored quantitatively and qualitatively. Specifically, we establish a monotonic relationship between the decay rates of queueing quantities and the correlation coefficient of the interarrival and service times. We also show that the decay rates are increasing in the correlation coefficient. In addition, queues with the maximum/minimum correlation coefficient are analyzed. Two examples are presented to demonstrate the importance of using the correlation coefficient in queueing performance/economic analysis.

Methods for radical reduction of packet loss probability in telecommunication networks with fractal traffic are developed. We investigate ways of preventing the losses within the framework of queueing theory; relevant simulation experiments are carried out. It is determined that strategy for the channel number increase in the network nodes has principally higher efficiency than that for the buffer increasing and/or channel performance increasing. Approximation methods for loss probability in the nodes of multiserver queueing system without buffers are investigated. The paper offers to approximate the loss probability in the node with n channels by steady-state probability in the state n of relating infinite-server queueing systems. We develop an analytical-statistical technique of optimal channel distribution over the nodes in networks with fractal traffic which is based on such approximation. The example of the method application is provided. The developed method could be used by engineers designing the telecommunication networks.

This paper investigates a multiprocessor queueing system in which each incoming task requires a variable number of service nodes, reflecting the diverse resource demands found in modern parallel computing environments. By modelling the task arrival and service processes using Poisson and exponential distributions, the system’s behaviour is analyzed through differentialdifference equations that govern the steady-state probabilities. The research provides a comprehensive solution for determining the probability of different numbers of tasks being serviced, considering both the acceptance and rejection of tasks based on system capacity. The derived results offer valuable insights into optimizing system performance in terms of throughput, task completion, and resource utilization for multiprocessor systems.

In this paper, the transient solution of a Markovian queuing system with two heterogeneous servers and retention of reneging customers is obtained. The explicit transient probabilities of system size are obtained using probability generating function technique. The corresponding steady-state probabilities are also derived. Further, the time-dependent mean and variance are obtained. Finally, a numerical example is provided to study the behavior of the system. The numerical results show that the heterogeneous system performs better than its homogeneous counterpart.

An exact analysis of the discrete-time single-server SMP/D/1/s queue with an arbitrary semi-Markov input process and a buffer size s is presented. The SSMP/D/1/s queue and the DMAP/D/1/s queue re covered in this model. The analysis yields easy expressions for the queue length density at arrivals (loss probability) and the waiting time density. This queuing system is suitable for modeling correlated input streams of asynchronous transfer mode (ATM) networks involving batches. Calculations show the influence of correlation on both loss probability and waiting time. >

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained.

Passage-time computation and aggregation strategies for large semi-Markov processes

When inter-arrival times between events follow an exponential distribution, this implies a Poisson frequency of events, as both models assume events occur independently and at a constant average rate. However, these assumptions are often violated in real-insurance applications. When the rate at which events occur changes over time, the exponential distribution becomes unsuitable. In this paper, we study the distribution of inter-arrival times of severe storms, which exhibit substantial variability, violating the assumption of a constant average rate. A new approach is proposed for modeling severe storm recurrence patterns using a finite mixture of log-normal distributions. This approach effectively captures both frequent, closely spaced storm events and extended quiet periods, addressing the inherent variability in inter-event durations. Parameter estimation is performed using the Expectation–Maximization algorithm, with model selection validated via the Bayesian information criterion (BIC). To complement the parametric approach, Kaplan–Meier survival analysis was employed to provide non-parametric insights into storm-free intervals. Additionally, a simulation-based framework estimates storm recurrence probabilities and assesses financial risks through probable maximum loss (PML) calculations. The proposed methodology is applied to the Billion-Dollar Weather and Climate Disasters dataset, compiled by the U.S. National Oceanic and Atmospheric Administration (NOAA). The results demonstrate the model’s effectiveness in predicting severe storm recurrence intervals, offering valuable tools for managing risk in the property and casualty insurance industry.

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.