Imbedded Markov Chain Models

This research aims to determine the queue model on a motorcycle parking area at Jember Universityand its performance measures. The data collection method in this research uses the observation method. Data taken are the number of arrivals, the number of departures, and service time. To get arrival distribution and service time distribution use mathematical calculation and Kolmogorov-Smirnov Test with SPSS Statistics. After arrival distribution and service time distribution are known, then using Kendall Notation rule to get queue model. Formula derivation and POM-QM for Windows are used to obtain performance measurement values in the queue model. After getting the data needed through observation, then analysedtaht data to determine the value of arrival rate and service rate, then the data is reanalysed to find out the data obtained has been steady-state. This research is known that steady-state, so that distribution test can be done, with the result that the arrival distribution was Poisson distribution and the service time distribution was Exponential distribution. In modelling the queue Kendall Notation is required. It’s used to clarify how the queue system reviewed works and is modelled. Based on the results of data analysis in this research, the queue model obtained was (M/M/1):(FIFO/∞/∞).

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

In this paper, we analyze a two station tandem queue with Poisson arrivals and exponential service times. All arrivals occur at the first line, and, after receiving service at the first station, they proceed to the second line. There is only a finite buffer between the stations, and, as soon as the buffer is full, any job completed by the first server is lost. To reduce customer loss, the first server can move to the second station and help the second server, thereby increasing its service rate. Once the work at station two is complete, the job leaves the system. The problem will be solved by using eigenvalues which can be obtained in explicit form. It is shown that this method is substantially faster than matrix analytic methods.

This paper presents an analytical study of a two-stage production system with provision for storage of intermediate product. Strict limits are placed on the size of the intermediate product inventory, and on the rates at which product can be moved into or out of inventory. Exponential service times are assumed at stage one and Erlang distributed service times are assumed at stage two. The problem is solved by formulation as a Markov process, and the impact of the inventory constraints on the production rate of the line is investigated. The problem of balancing the system is considered at some length.

An arbitrary configuration of an open queueing network with exponential service times and finite buffers is analyzed. We offer an iterative procedure for approximating the marginal occupancy probabilities for each queue of the system. The method decomposes the queueing network into individual queues and analyzes each in isolation using information from only its nearest neighbors. Based upon the SIMP approximation previously used for tandem queues, it replaces each server's service time with a clearance time, which includes blocking, and each server's arrival rate by an equivalent acceptance rate. The procedure is easy to implement and requires modest memory and computer time. Extensive numerical experiments, performed for various topologies, yield accurate results compared with those obtained by exact or simulation methods.

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.

Mathematical models of the average production rates are derived for two and three-stage production lines. The service times at each of the stages are assumed to be exponential random variables. An investigation is made of the effect of the amount of internal storage capacity on the output rate of the line. Additional investigations involve the optimum allocation of storage capacity and the decrease in the production rate caused by adding stages to the line.

Mathematical models of the average production rates are derived for two and three-stage production lines. The service times at each of the stages are assumed to be exponential random variables. An investigation is made of the effect of the amount of internal storage capacity on the output rate of the line. Additional investigations involve the optimum allocation of storage capacity and the decrease in the production rate caused by adding stages to the line.

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product formsolution for the network. This paper extends previous results to arbitrary service with finite mean (the nondifferentiable case included). This is done by introducing supplementary variables for the remaining service requirements at the various queues in the system. By using generalized function (in the sense of Schwartz) a new technique is presented for handling queueing networks in which service distributions may be general.

We examine multiple independent sources providing status updates to a monitor through simple queues. We formulate an age of information (AoI) timeliness metric and derive a general result for the AoI that is applicable to a wide variety of multiple source service systems. For first-come first-served and two types of last-come first-served systems with Poisson arrivals and exponential service times, we find the region of feasible average status ages for multiple updating sources. We then use these results to characterize how a service facility can be shared among multiple updating sources. A new simplified technique for evaluating the AoI in finite-state continuous-time queuing systems is also derived. Based on stochastic hybrid systems, this method makes AoI evaluation to be comparable in complexity to finding the stationary distribution of a finite-state Markov chain.

This paper is concerned with the Bayesian analysis of general queues with Poisson input and exponential service times. Joint posterior distribution of the arrival rate and the individual service rate is obtained from a sample consisting inn observations of the interarrival process andm complete service times. Posterior distribution of traffic intensity inM/M/c is also obtained and the statistical analysis of the ergodic condition from a decision point of view is discussed.

Formulating the solution as an optimisation problem has proven to be effective in developing solutions to many real world problems. We generally obtain the best possible solution using these methods. In this work, the traffic scheduling problem has been formulated as a waiting time minimisation problem, and appropriate cost functions have been developed, in pursuit of finding the optimal solution. A first-in, first-out queuing model is used, with the vehicles arriving in a Poisson process, and the service time being exponentially distributed. The key feature of this model is that it adapts to varying service and arrival rates of the lanes. These rates are forecast using a neural network model, and appear in the objective function. It was observed that the use of the neural network greatly improved the robustness of the model. Although the model has been developed for a four lane, two way architecture, it can be generalised to any architecture. Results have been analysed by comparing the proposed method to a proportional time distribution. It is shown that the proposed model performs relatively well, when there is rapid variation in the arrival and service rates.

In this paper, we consider a continuous review perishable inventory system with multi-server service facility. In such systems the demanded item is delivered to the customer only after performing some service, such as assembly of parts or installation, etc. Compared to most inventory models in which inventory is depleted at the demand rate, it is depleted, in this model, at the rate that the service occurs. We assume that the arrivals of customers are according to a Markovian arrival process (MAP) and that service time has exponential distribution. The ordering policy is based on (s; S) policy. The lead time is assumed to have exponential distribution. The customer who finds either all servers are busy or no item (excluding those in service) is in the stock, enters into an orbit of infinite size. These orbiting customers send requests at random time points for possible selection of their demands. The interval time between two successive request-time points is assumed to have exponential distribution. In addition to the regular customers, a second flow of negative customers following an independent MAP is also considered which will remove one of the customers in the orbit. The joint probability distribution of the number of busy servers, the inventory level and the number of customers in the orbit is obtained in the steady state. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated.

A multi-server perishable inventory system with negative customer

We iteratively derive the product‐form solutions of stationary distributions for a type of preemptive priority multiclass queueing networks with multiserver stations. This type of queueing systems can typically be used to model the stochastic dynamics of some large scale backbone networks with multiprocessor shared‐memory switches or local (edge) cloud computing centers with parallel‐server pools. The queueing networks are Markovian with exponential interarrival and service time distributions. The obtained iterative solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies. Numerical comparisons with existing Brownian approximating model (BAM) related to general interarrival and service times are provided to show the effectiveness of our current designed algorithm and our previous derived BAM. Furthermore, based on the iterative solutions, we can also give some analysis concerning network stability for some cases of these queueing systems, which provides some insight for more general study.