Optimality of threshold policies in single-server queueing systems with server vacations

In this system we consider a batch arrival Poisson input with two phases of heterogeneous service with random feedback in each servivce. The first phase of service is essential for all customers, but with a probability tagged customer chose second phase, feedback to tail of original queue or leave the system. Also, after completion of the second phase, with a probability the customer leaves the system, or feedback to tail of original queue. At each service completion epoch, the server may apt to take a vacation with a probability or continue to be available in the system for the next service. The service times are assumed to be general. The vacation period of the server has two heterogeneous phases with Coxian-2 distribution. The vacation times are assumed to be general. When the server goes for vacation, service become unavailable and customers may decide to renege at each vacation times. We assume reneging follow exponential distribution. All stochastic processes involved in this system (service and vacation times) are independent of each other. We derive the PGF’s of the system and by using them the performance measures are obtained. Some numerical approch are examined the validity of results.

This paper presents a discrete time bulk service queue under the policy (a, c, d) with state dependent service rates. The inter-arrival times are assumed to be independent and geometrically distributed. The customers are served by a single server under the policy (a, c, d) with state dependent service rates. In this model the server begins service only when there are at least c units in the queue and he serves a maximum of d units in a batch. The server continue to serve even when the queue size is less than c but not less than a secondary limit a after a service completion epoch, but with a different service rate. If after a service completion epoch the queue size is less than a, the server becomes idle. The service times are also assumed to be independent and geometrically distributed. The steady state probabilities and some performance measures of the model are computed.

Tandem queues are good mathematical models for description of information transmission in various communication systems and networks. These queues play also an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. So, their investigation is interesting for theory and applications. In this paper, we consider tandem queue suitable for modeling the systems and networks where information flows are correlated and bursty what is typical for many modern telecommunication networks. Possible correlation of customers inter-arrival times and batch arrivals are taken into account via of consideration of the Batch Markovian Arrival Process (BMAP) as input stream to the system. The system consists of two stations. The service time at the station 1 is assumed to be generally distributed. There is no buffer at this station, and customers who meet the busy server repeat attempts to enter the system in random time intervals. The service process at the station 2 is assumed to be described by the continuous time Markov chain with a finite state space. This assumption holds good, e.g., if the station 2 has a finite buffer, consists of a finite number of identical or heterogeneous servers where the service time distribution is assumed to be of PH (PHase) type. Markov chain embedded at service completion epochs at the station 1 and the process of system states at arbitrary time are under study. Ergodicity condition and algorithms for computing the steady state probabilities are presented.

A system of two parallel queues is considered, where each customer must leave after service through a common gate G. It is assumed that service times at the two stations I and II are independent and identically distributed, and that exit service takes a fixed length of time. A I-customer may be served at station I only if the previous I-customer has completed exit service. Integral equations are formulated from which the distribution of the total service time may be obtained when the two queue sizes are infinite. These equations are solved for exponential and generalized erlangian service times. Extensions to the case of k parallel queues and to the case of Poisson arrivals and finite queue sizes are discussed briefly.

Quality of Service (QoS) is a critical factor for businesses, directly impacting customer satisfaction and competitiveness. At Ethio Telecom in Ethiopia, effective mechanisms for attracting and retaining impatient customers are essential. Additionally , incorporating synchronous or asynchronous server vacations enhances the realism of queuing-inventory systems. Multi-server vacation models offer greater flexibility and applicability than single-server systems. This paper examines a finite-capacity, multi-server, Markovian queuing-inventory system (MQIS) with attraction-retention mechanisms for impatient customers and C-removable servers under an asynchronous vacation policy. In this model, D (0 < D < C) servers take a vacation together when no customers are waiting at a service completion epoch, while the remaining (C − D) servers either continue serving (if inventory permits) or remain idle (if inventory is depleted). Service and vacation times follow independent exponential distributions, and inventory is managed with a (0, q, Q) ordering policy, which triggers an order to replenish inventory to Q when levels drop to q (q < Q). Using Markov process theory, we develop the queuing-inventory model, derive its steady-state solutions, and compute various performance metrics. Additionally, we propose a cost-loss model to determine the optimal service rate and the optimal number of vacationing servers, applying a genetic algorithm for cost analysis. Finally, we explore how system parameters influence the optimal number of vacationing servers, inventory policy, and cost-loss value. The model is illustrated using data from Ethio Telecom?s service center in Arba Minch City, Ethiopia, with potential applications in supermarkets , telecommunication centers, and restaurants. Future research may examine queuing-inventory systems with batch services, retrial customers, and warehouse disruptions.

In this paper, we consider two queueing models. Model I is on a single-server queueing system in which the arrival process follows MAP with representation D = (D0,D1) of order m and service time follows phase-type distribution (β, S) of order n. When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has k stages. The pth stage parameter is θp for 1 ≤ p ≤ k. If a customer completes the service in between the realizations of stages k1 and k2 (1 < k1 < k2 < k) of the clock, it is a perfect one. On the other hand, if the service gets completed either before the kth1 stage realization or after the kth2 stage realization, it is discarded because of imperfection. We analyse this model using the matrix-geometric method. We obtain the expected service time and expected waiting time of a tagged customer. Additional performance measures are also computed. We construct a revenue function and numerically analyse it. In Model II, a single server queueing system in which all assumptions are the same as in Model I except the assumption on service time, is considered. Up to stage k1 service time follows phase-type distribution (α′ , T′) of order n1 and beyond stage k1, the service time follows phase type distribution (β′ , S′) of order n2. We compare the values of the revenue function of the two models

The proposed model deals with the study of two server retrial queueing system with vacation times and discouraged customers. The servers under consideration are functioning with variant speeds to provide service. The customers will directly enter into the service channel if any one of the server is found free upon arrival. Otherwise, the customer will move to the orbit to occupy the server later. At the service completion epoch, the server turns to vacation state if no more customers were found in the waiting area. During this vacation or working period, the customers may leave the system forever due to their longer waiting time. This Markov model of birth-death process is developed using the Chapman-Kolmogorov governing equations. By using the recursive approach, we obtain the stationary distributions for the arriving customers. Few performance indices were identified to depict the behavior of the formulated model.

Single server queueing models are important in the study of a wide variety of stochastic systems. A particularly important example is the study of task schedules for computer systems with real-time applications. In this paper, we present a class of approximations for the waiting time distribution in single server queueing systems with general independent (renewal) input and general (independent) service time distributions. These approximations allow the analyst to use as much (or as little) of the structure of the input and service processes as desired. Moreover, they also allow him to concentrate on specific quantities associated with the delay distribution that may represent the most relevant performance criteria. Examples include probability of delay, mean delay, and tails of the delay distribution. The methods given in this paper have been used to analyze the performance of task schedules in a variety of processor-based systems.

In this article some approximations for the mean number in the single server queueing system are presented and compared. Based on numerical comparison a guide is provided to select the appropriate approximation(s) for different ranges, of the squared coefficient of variation of the inter-arrival and service time distributions. Two approximations for the distribution of the number in the system are given and the better is chosen.

We consider the optimal control problem of certain batch service queueing systems with compound Poisson arrivals and linear holding costs. The control problem involves the determination of the epochs at which the service is initiated as well as the sizes of the batches served. The service times are assumed to be independent and identically distributed, however, with a general distribution. A quite natural operating policy is to start the service as soon as the number of customers reaches some threshold and serve always as many customers as possible. Assuming infinite service capacity Deb [4] proved that under some mild conditions the optimal operating policy is of this type. In this paper we show that a similar result is valid even if the service capacity is finite. In this case the threshold is never greater than Q, the service capacity (the maximum number of customers that can be served at the same time).

We study a vacation queueing system with a single server simultaneously dealing with an M/G/1 and an M/D/1 queue. Two classes of units, priority and non-priority, arrive at the system in two independent Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority units and a deterministic service to the non-priority units. We further assume that the server may take a vacation of random length just after serving the last priority unit present in the system. We obtain steady state queue size distribution at a random epoch. Corresponding results for some special cases, including the known results of the M/G/1 and the M/D/1 queues, have been derived.

A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in theMx/GI/1/kqueue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensityρ< 1 and for the virtual waiting time process inM/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality ofGI/MY/1/ktoMx/GI/1/k +1, we get a similar result for the system queue length distribution just before the arrival of a customer inGI/MY/1/k.We apply those results to prove that the loss probabilities ofMx/GI/1/kandGI/MY/1/kare increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.

A finite-buffer queueing model is considered with batch Poisson input and controllable service rate. A batch that upon arrival does not fit in the unoccupied places of the buffer is partially rejected. A decision to change the service mode can be made at service completion epochs only, and vacation (switch-over) times are involved in preparing the new mode. During a switch-over time, service is disabled. For the control of this model, three optimization criteria are considered: the average number of jobs in the buffer, the fraction of lost jobs, and the fraction of batches not fully accepted. Using Markov decision theory, the optimal switching policy can be determined for any of these criteria by the value-iteration algorithm. In the calculation of the expected one-step costs and the transition probabilities, an essential role is played by the discrete fast Fourier transform.

A single-server queueing system with bounded buffer, Poisson flow, and gamma service time distribution is studied. Limiting values of certain performance indexes for unboundedly increasing service time variation coefficient are determined and approximate formulas for the performance characteristics of the system for large variation coefficients are derived.