In this paper, we analyze a non-Markovian queueing model providing service in up to K sequential phases. After completing each service phase, a customer either proceeds to the next with probability q_i or exits the system with probability q_i=(1-q_i) . The model incorporates the possibility of server breakdowns, where the server may fail randomly and is then sent for repair. Customers arrive in batches of random size according to a Poisson process. If the server is busy upon arrival, they join a retrial group and attempt re-entry after a random time. Customers are impatient and may balk if the server is busy or under repair. Additionally, once the system becomes empty, the server takes a vacation; if still idle upon return, it may take one additional optional vacation selected from among m available options. The Supplementary Variable Technique (SVT) is employed to derive system performance measures, and numerical experiments are conducted to perform sensitivity analysis.

In the present article, we investigate a queuing system in which units arrive in random batch sizes according to a Poisson process, while service is provided to individual units on a one-by-one basis. The service mechanism comprises a mandatory service phase followed by an optional service phase, undertaken according to the unit’s requirements. The server may stop functioning as a result of an unpredictable breakdown while offering any one of the services. Once the service is interrupted due to a server failure, the initiation of repair activities may be delayed owing to the unavailability of repair personnel or the required repair facilities. Following this delay, the server undergoes multiple essential repair phases before being restored to an operative state. The current non-Markovian model is investigated by employing a supplementary variable approach, and the time-independent behavior and different performance parameters of the system are discussed by means of a probability-generating function. By defining the suitable parameters, some exceptional cases are also described. Finally, the validity of the obtained analytical results is demonstrated through a numerical example.

The present paper deals with multi-server Markovian queuing model with impatient behaviour of units.The server work in two modes: slow mode and fast mode; and changes its state from slow to fast or vice versa with some exponential distributed time parameters.When the server works in slow mode, the units may leave the system due to impatient behaviour of units.We investigate the present model by two variant, one with finite capacity and other with finite population.Transient probabilities of various states are obtained by solving the set of governing equation via Runge-Kutta method.The various system performance measures are established in terms of transient probability of the system states.The system behaviour for different parameter are analysed by using the graphs and tables.

This paper deals with MX/G/1 queueing system in which arriving units join a single waiting line. Server provides the first essential service and one of the optional services among m available optional services, to all arriving units. After completion of both phases of services of each unit the server may take optional vacation with probability p. It is assumed that during any phase of service, server may stop working due to random failure and is sent for repair. Further it is assumed that arriving units may balk from the system when server is busy, vacation and under repair with probability -b = 1 – b. Using the probability generating functions we derive the queue size distribution at different time points as well as waiting time distribution. Finally numerical illustration is provided to analyse the sensitivity of different parameters on various performance measures.

The present study deals with the performance analysis of a single server queues with state dependent arrival rates in batches. It is considerable that the units join the system for essential and optional services available in the system. The server can avail the at most m-vacations under randomised vacations policy. The system is analysed by using the supplementary variable technique with the assumptions of probability reasoning for study the steady state behaviour of the system and to obtain various performance indices of queue size distribution. The maximum entropy principle is used to find the approximate results of performance measures and comparative study of exact and approximate waiting times of the units is discussed. The numerical illustration is also carried out to facilitate the minimum cost for optimal parameters and validate the other results of the system.

The present investigation deals with a stochastic model of queueing system wherein the customers join the system in bulk with different arrival rates. On arrival in the system, the customer may choose any of the available m-optional services and the server may avail the optional vacation after finishing any one of the service or stay in the system to provide the service to other customers. It is also assumed that the server starts the service, if at least N customers are there in the queue. The steady state behaviour of the system is discussed by using the supplementary variable technique. The mathematical model is analysed by using the probability generating functions of queue size distribution to obtain the performance indices of the system. The principle of maximum entropy is also employed to find the approximate values of waiting time of the system. To validate the results, the numerical illustration is presented.

The present investigation deals with the analysis of Markovian queueing model with finite capacity and finite population wherein the server works in fast and slow random environments, depending on the status of service system, with exponential distributed time parameters. On arrival of the customer in slow mode, the exponential distributed time may be fixed. Due to the impatient behaviour (reneging), the customer may leave the queue after some time of joining the system if the server does not change its state before expiry of the timer. To discuss the behaviour of the queue length distribution and to obtain the transient solution, Runge-Kutta method of the fourth order is used. Various performance measures are determined in terms of transient probabilities of the system states. The numerical illustrations are facilitated to validate the tractability of performance measures as far as computational aspect is concerned.

In this paper we study some basic concept of queuing theory and provide brief overview of queuing theory.We analyze the basic component of queuing theory and different type of distribution that are used to analyze a queuing model.The importance and requirement of state dependent queuing model also explained.Finally some basic model of queuing theory, performance measures is discussed and methodologies used to analyze such model are explained.

In this paper, retrial queue with unreliable server and bulk arrivals is investigated. The server is capable of providing m-optional services and any one of these available services, may be rendered to the customer after the first essential service if the customer opts for the same. It is assumed that the server may fail while rendering any phase of service and undergoes for the immediate repair. After the completion of the service of a customer, the server may either take a vacation for a random period or may continue to provide the service to the other customers waiting in the queue. The supplementary variables corresponding to service time, repair time and retrial time are incorporated to determine the queue size distribution. To examine the effect of different parameters on the performance measures of the system, the numerical illustration is given which is supported by numerical simulation and sensitivity analysis.

In this paper, a single server queueing model, wherein the units arrive in bulk with varying arrival rates in Poisson process, is considered. It is assumed that the service time of units is arbitrarily distributed. Also, we incorporate the optional deterministic vacations for the server. The server may take a vacation of a fixed duration at the completion of each service or may continue to be available in the system for the next service. At busy and vacation states, the customers may balk from the system with different balking probabilities. By using the basic assumptions of probability reasoning and supplementary variable technique, the steady state behaviour of the system is studied and various performance measures are obtained. In order to obtain the approximate values of the system state probabilities, the principle of maximum entropy is also employed. To verify the tractability of the performance measures obtained, the numerical illustrations are provided. Further, the sensitivity analysis is carried out to examine the system performance with respect to different parameters.

The present investigation deals with the single server stochastic model of queueing system, wherein the arrivals of units are in batches and follow Poisson process with state dependent arrival rates. There is a provision of two stages of heterogeneous service with arbitrary distributed different service time. The server may take optional vacation after the completion of the both stages of service of each unit. The server may fail at any instant of service and requires repair. The transient and steady state behaviours of the queue length distribution are studied by using the Laplace transform and probability generating function along with supplementary variable-based methodology, to obtain the performance measures of the system. Some particular cases are discussed by setting the parameters. The maximum entropy principle is applied to find the approximate system state probabilities. The numerical illustrations are also considered to validate the analytical results.

The present paper deals with a single server queueing system wherein the customers join the system in bulk with varying arrival rates. The first stage regular service and second stage optional service are provided to the customers. The server may breakdown at any instant during any stage of service. There is a provision of essential and optional repair in m phases. To discuss the steady state behaviour of the system, the assumptions of probability reasoning and supplementary variable technique is employed. Various performance indices are obtained using the generating function approach. To verify the validity of these indices, the numerical illustrations are provided. The technique of principle of maximum entropy is also considered to find the approximate values of the system probabilities for the queue length distribution.

In this paper, we investigate a single-server Poisson input queueing model, wherein arrivals of units are in bulk. The arrival rate of the units is state dependent, and service time is arbitrary distributed. It is also assumed that the system is subject to breakdown, and the failed server immediately joins the repair facility which takes constant duration to repair the server. By using supplementary variable technique, we obtain the probability generating function of the number of units in the system which is further used to establish some performance indices such as the mean number of units in the system, mean waiting time, etc. Special cases are also discussed. In order to obtain approximate values of system state probabilities, the principle of maximum entropy is employed. Numerical results are also presented to validate the analytical formulae.

This investigation deals with single server queueing system wherein the arrival of the units follow Poisson process with varying arrival rates in different states and the service time of the units is arbitrary (general) distributed. The server may take a vacation of a fixed duration or may continue to be available in the system for next service. Using the probability argument, we construct the set of steady state equations by introducing the supplementary variable corresponding to elapsed service time. Then, we obtain the probability generating function of the units present in the system. Various performance indices, such as expected number of units in the queue and in the system, average waiting time, etc., are obtained explicitly. Some special cases are also deduced by setting the appropriate parameter values. The numerical illustrations are provided to carry out the sensitivity analysis in order to explore the effect of different parameters on the system performance measures.

This investigation deals with single server state dependent queuing systems, wherein the arrivals of units are in batches and follow the Poisson process with state dependent arrival rates. After availing of the First Regular Vacation (FRV) in a case when there is no customer in the system, the server may also take a Second Optional Vacation (SOV). By using supplementary variable techniques, the probability generating function of the queue length distribution is established to study various performance measures. The maximum entropy approach is also used to find queue length distribution for evaluation of steady state probabilities in all different states. Numerical illustrations are provided to verify the tractability of performance measures obtained analytically.

This paper investigates a single server bulk queueing system with state-dependent rates and second optional service. The service time of the essential service is general-distributed whereas that of optional service follows the exponential distribution. By using supplementary variable technique, the probability-generating functions of the queue length distribution have been obtained. Further, the queue length distribution is established by using maximum entropy approach. The numerical illustrations are provided to verify the tractability of performance measures. The sensitivity analysis is also carried out to examine the effect of system descriptors.

This investigation deals with single server markovian queueing model with controllable arrival rates with discouragement factor reneging, in which it is assumed that the arrival and service processes are interdependent. The stationary state solutions of the model are analysed. The expressions for system characteristics as average number of customers in the system and average waiting time are determined. The numerical illustrations are also considered to validate the analytical results and to illustrate the effect of the parameters on several performance characteristics.