Optimal Control of an M/Ek/1 Queueing System with a Removable Service Station

The article considers the components of the target function of optimization of HUNDRED losses associated with the introduction of the post (posts) of acceptance-diagnosis with the increase of the set of diagnostic equipment. In particular, the target function of optimizing losses at the service station is presented in the form of an economic criterion that takes into account the maximum possible number of components on which depends the technological efficiency of the service station and the level of "satisfaction" of car owners with services. The object of research is a car service station. The purpose of the work is to conduct a study of the target function of optimization of HUNDRED losses associated with the type of organization of acceptance-diagnosis of cars and to find out algorithms for finding the components of the target function based on the representation of HUNDRED as a queuing system. Research method - analysis and formalization: quantitative indicators of the objective function, methods of forming a group of events, alternative solutions and calculation of the main components of optimization of HUNDRED losses as a queuing system. It is established that the target function of optimization of HUNDRED losses has economic essence and can be presented in the form of complex criterion. It is found that based on the theory of queuing for the area of reception-diagnosis it becomes possible to find solutions of the system of equations that describe its current state, namely: 1) the probability that the system has no requirements, ie free all "workers »Posts and no applications in the queue; 2) the probability of finding the system in a certain state; 3) the probability of denial of the received application for maintenance, ie the probability of finding the system in the last valid state; 4) the average number of occupied reception posts. Similar solutions were found for the TO-R zone. The formulas for determining the components of the duration of the application are considered and given. The formulas for determining the average value of the duration of work in the area of maintenance by the flow of faults and their share in the total flow of applications. The necessity of the 5th - 6th step of building the equipment in the reception-diagnostic area for detection of additional malfunctions (hidden defects) is substantiated. It is established that the average processing time of the application at the posts of acceptance-diagnosis includes three components: 1) the average time of acceptance-diagnosis of the declared ambiguous manifestation of the fault; 2) the average time of additional preliminary diagnosis on the application, which contained ambiguous manifestations of malfunctions. Depends on the number of diagnostic operations performed and the available equipment; 3) the average time of additional diagnosis on the application, which either did not contain ambiguous manifestations of malfunctions, or they did not fall within the capabilities of the existing diagnostic complex at the reception. The results of the article can be used to model the operation of the service station as a queuing system in computer software products and will reveal the low efficiency of diagnosing the main systems of cars in a particular service station. KEY WORDS: DIAGNOSTIC PARAMETER, APPLICATION, PROBABILITY, FAULT, EQUIPMENT, POST, PROFIT, ACCEPTANCE-DIAGNOSIS, SYSTEM.

Applications of Queuing Theory in the Tobacco Supply

Reliability analysis of M/G/1 queueing system with repairable service station of reliability series structure

This paper studies the optimal control of an M[x]/G/1 queue with two types of generally distributed random vacation: type 1 (long) and type 2 (short) vacations. The server is turned off and takes type 1 vacation whenever the system is empty. If the number of customers waiting in the system at the instant of a vacation completion is less than Q, the server will take a type 1 vacation. If the number of customers in the system is greater than or equal to Q and smaller than N, the server will take a type 2 vacation. If the server returns from a vacation and finds at least N customers in the system, he is immediately turned on and requires a startup time before providing the service until the system is again empty. We analyze the system characteristics for such a model. The total expected cost function per unit time is developed to determine the optimal thresholds of Q and N at a minimum cost.

The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns☆

In modern methods of technological design, the capacity of service facilities depends on the number of cars entering them per unit of time. At the same time, regardless of the type of service object, the number of visits is somehow associated with intensity of the vehicle traffic on a given road, as well as with one of the characteristics that determine speed of this event. Such a characteristic, for example, for service or maintenance stations is intensity of vehicle failures, for fuelling stations – intensity of fuel consumption, for hotels and parking lots – intensity of the onset of fatigue for drivers and passengers. This gives ground for conclusion that there is a significant similarity between the processes that result in an event that can be called a visit to a service facility. Thus, the purpose of this theoretical study was to develop a common approach applicable to all road service facilities to determine the number of such events, as well as to develop a common methodology for their design.A characteristic feature of operation of service objects is a time-varying flow of demand for the operation of service facilities, as well as variable labour intensity and duration of troubleshooting or the provision of other service services. Systems in which the moments of receipt of requests for service and the duration of the services themselves are variable and random are called a queuing system. Therefore, to justify the number of car arrivals at road service facilities, the authors proposed a probabilistic method, the main tools of which are provisions and mathematical apparatus of the queuing theory.The main condition for functioning of the queuing system is the ratio between the incoming flow of requirements and the absolute throughput of the system. Thus, the mathematical model proposed by the authors considers the performance indicators of service facilities, depending on the size of the incoming flow of requirements and its variation, on the one hand, and on the throughput and performance of service facilities, on the other hand.

As the IEEE 802.16e Wireless Metropolitan Access Network (WMAN) supports the mobility of a mobile station (MS), increasing MS power efficiency has become an important issue. In this paper, we analyze the sleep-mode operation for an efficient power saving mechanism for delay-guaranteed services in the IEEE 802.16e WMAN and observe the effects of the operating parameters related to this operation. For the analysis we use the M/GI/1/K queueing system with multiple vacations, exhaustive services and setup times. In the analysis, we consider the power consumption during the wake-mode period as well as the sleep-mode period. As a performance measure for the power consumption, we propose the power consumption per unit time per effective arrival which considers the power consumption and the packet blocking probability simultaneously. In addition, since we consider delay-guaranteed services, the average packet response delay is also considered as a performance measure. Based on the performance measures, we obtain the optimal sleep-mode operation which minimizes the power consumption per unit time per effective arrival with a given delay requirement. Numerical studies are also provided to investigate the system performance and to show how to achieve our objective.

We analyzed an M/M/R queueing system with finite capacity N where customers have multiple classes with no priorities under steady-state conditions. It is assumed that any one class of arriving customer is serviced by one or more servers, and that each customer class has equal probability of service. Analytic explicit solutions are obtained with the assumption of exponential distributions for arrival times and service times. A cost model is developed to determine the optimal system capacity and the optimal number of servers. The minimum expected cost, the optimal system capacity, the optimal number of servers, and various system performance measures are obtained for three customer classes and four customer classes, based on assumed numerical values given to the designated system parameters.

This paper studies the optimal control of an N-policy two-phase MX/Ek/1 queueing system without gating and with server startup and breakdowns. Explicit expressions for the steady state distribution of the number of customers in the system are obtained and also derived various system measures. A cost model is formulated to determine the optimal operating policy at a minimum cost. Sensitivity analysis is performed through numerical illustrations.

A tandem retrial queueing system with two Markovian flows and reservation of channels

We consider a finite-capacity single-server queuing system in which the problems of capacity design and service quality control are integrated. The amount of time spent on servicing a customer is used as a measure of the quality of service provided. The expected reward from servicing is assumed to be a nondecreasing function of the service duration, while a linear holding cost is assumed for customers waiting in the system. We maximize the long-run average return per unit time by optimally controlling the quality of service to be provided as a function of the workload facing the server for different waiting-room capacities. We show that the optimal control policy for a given capacity is monotone and also examine the effect of varying the system capacity on the optimal control policy. Furthermore, we analyze the design problem of selecting an optimal capacity by assuming that an optimal service control policy will be followed for each system capacity.

In this paper we study the problem of optimal controlling in a processor sharing (PS) M/M/2 queueing system with heterogeneous servers. The servers differ in the service intensities, operating and usage costs. The objective is to find the optimal policy to allocate the customers either to an idle or partially loaded server, or to the queue at each arrival and service completion epoch to minimize the long-run average cost per unit of time. We handle this optimization problem as Markov decision problem and study numerically structural properties of the optimal control policy. Using a policy-iteration algorithm we show that this policy for the current model is of threshold type. In this case the faster server handles customers with maximum capacity, while the number of simultaneously serviced customers at the slower server can be increased only when the number of waiting customers exceeds a certain threshold level. The data-sets generated by classic methodology of analyzing the controlled queues are used to explore predictions for optimal thresholds through artificial neural networks. The presented theoretical results are accompanied by heuristic solution and numerical examples.

We consider the economic behavior of a single-channel queuing system operating with the following cost structure: a server start-up cost, a server shut-down cost, a cost per unit time when the server is turned off, a cost per unit time when the server is turned on, and a holding cost per unit time spent in the system for each customer. For the undiscounted, infinite-horizon problem, bounds are obtained for the cost rate and optimal policy when the interarrival time distribution is a member of the class of increasing-failure-rate distributions. Some of the bounds hold for all interarrival-time distributions.

We consider queueing systems in which the service rate is the controlled variable. The cost depends on the queue length and selected rate. The objective is to choose the service rate dynamically, based on the state of the system so as to minimize the average cost over an infinite horizon. Six classes, either known in the literature or new, are studied in detail: queueing systems with vacations, single-server queueing systems with and without switching costs, and tandem queueing systems with and without service costs. A novel approach is presented here using fuzzy control to solve these problems. Simulation shows that the approach is efficient and promising, especially in cases where analytical solutions do not exist.

This dissertation examines an M[x]/G/1 queueing system with a randomized vacation policy and at most J vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1-p. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the Jth vacation, the server becomes idle in the system until at least one customer waiting in the queue. In this dissertation, we investigate the following three queueing systems: Reliable server queueing system, un-reliable server queueing system and un-reliable server with a delayed repair queueing system. For the three systems considered in our dissertation, using the supplementary technique, we develop the system size distribution as well as other important system characteristics, such as the system size distribution at busy period initiation epoch, the queue size distribution at a departure epoch, and the distributions of busy period and idle period, etc. Further, for the un-reliable server we also develop main reliability indices of the presented model. Using the renewal reward theorem, a cost model is constructed to determine the optimal randomized vacation policy. Based on the cost model, a heuristic approach is provided to search the joint optimum values of p and J. Some numerical results are presented for illustrative purpose. Our study presents an extension of the existing vacation queueing model and the analysis of the proposed model will provide a useful performance evaluation tool for more general situations arising in real word.