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In this paper, we analyse a single server preemptive priority queue with phase-type vacation and repair, feedback, working breakdown, close-down and impatient customers. Customers arrive according to the Marked Markovian Arrival Process and their service time according to Phase-type distribution. If the High Priority customers need feedback, they lose their priority and join the Low Priority queue. At any instant, if the server is broken down, the server provide service with slow mode for that current customer and then the server will go into a repair process. When there are no customers present in both the queues, the server close-down the system and then goes on vacation. During the close-down and vacation period, high priority customers may balk. The Matrix Analytic Method is used to look into the number of consumers that are currently in the system. Analysis of the steady-state, the server active period, and the total cost are all discussed. Finally, some significant performance measures and numerical examples are given. [ABSTRACT FROM AUTHOR]

The impact of information about the quality of service on marketing can be demonstrated by a competition between an observable queue and an unobservable queue. An illustration was given by an earlier paper of Hassin (Manag Sci 42:618–623, 1996), which considers two gas stations located one after the other on a main road. A driver who needs to fill the gas can see the queue length upon arriving at the first station, but not the second one. The driver decides which station to enter based only on this partial information for minimizing the expected waiting time. Via extensive numerical experiments, Hassin showed that the first server enjoys a larger market share, which led him to making a number of challenging conjectures. In this paper, we prove that all the numerical observations and conjectures are true so that the advantage of being the first server is assured. [ABSTRACT FROM AUTHOR]

Cloud Computing (CC) and Internet of Things (IoT) are upgrowing human intervention to enhance the daily lifestyle. Currently, the heavy loaded traffic congestion is a very big challenge over IoT-based applications. For that purpose, the researchers approached various ways to overcome the congestion mechanism in recent years. Even though, they have futile to acheive the best resource storage accessing capacity expectation other than, Cloud Computing. Data sharing is a key impediment of Cloud Computing as well as Internet of Things. These are the constituent that give rise to the combination of the IoT and cloud computing paradigm as IoT Cloud. Though, preserving the missed data during the execution time is a key factor to indulge the Retrial Queueing Theory (RQT), who is facing issue upon accessing Cloud Service Provider (CSP) enter into virtual pool to preserve the data for reuse. The paper imposes Markov Fluid analysis with Matrix Analytic Method (MAM) allows the data as continuous length of data rather than individual data to avoid the congestion. The virtual orbit queue follow constant retrial rate discipline, that is, head of the orbital users makes attempt to occupy the server are assumed to be independent and identically distributed (i.i.d). Steady-state expression presented to study the behaviour of congestion. An illustrative analysis is produced to gain deep perception into the system model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a queueing inventory system with batch arrival and batch service processes. Customers arrive in batches of sizes 1 , 2 , ... , k , according to a marked compound Poisson process. We call a batch of customers as belonging to j when there are j individual customers in that batch. The service facility has waiting rooms for each category of customers and also a room to serve them. Except for category 1, all other customers have finite waiting rooms. The service room has only a limited number of seats. These seats are arranged in such a fashion that customers belonging to category j have groups of seats, each with j seats for j = 1 , 2 , ... , k . Customers are taken for service according to the availability of seats designated to each category. A category j customer can be taken for service only if j items are available in the inventory. The service time of customers of category j is exponentially distributed with parameter depending on j for j = 1 , 2 , ... , k . The number of seats available in the service room for each category of customers is restricted to a finite number. The replenishment for items follows the (s , S) policy: fill up to S at the time of replenishment. Lead time follows an exponential distribution. We analyze the system in the equilibrium state. Performance characteristics are evaluated and a number of numerical illustrations are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider the batch arrival of customers to a transport station. Customers belonging to each category is considered as a single entity according to a BMMAP. An Erlang clock of order m starts ticking when the transport vessel reaches the station. When the Lth stage of the clock is reached, an order for the next vessel is placed. The lead time for arrival of the vessel follows exponential distribution. There are two types of rooms in this system: the waiting rooms and the service rooms for customers in the transport station and in the vessel, respectively. The waiting room capacity for customers of type 1 is infinite whereas those for customer of type 2 , ... , k are of finite capacities. The service room capacity C j for customer type j , j = 1 , 2 , ... , k is finite. Upon arrival, customers of category j occupy seats designated for that category in the vessel, provided there is at least one vacancy belonging to that category. The total number of vessels with the operator is h * . The service time of each vessel follows exponential distribution with parameter μ. Each group of customers belong to category j searches independently for customers of this category to mobilize passengers when the Erlang clock reaches L 1 where L 1 < L . The search time for customers of category j follows exponential distribution with parameter λ j. The stability condition is derived. Some performance measures are estimated. [ABSTRACT FROM AUTHOR]

We consider a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) { (X n , J n) } on Z + 2 × S 0 , where X n = (X 1 , n , X 2 , n) is the level state, J n the phase state (background state) and S 0 a finite set, and study asymptotic properties of the stationary tail distribution. The 2d-QBD process is an extension of usual one-dimensional QBD process. By using the matrix analytic method of the queueing theory and the complex analytic method, we obtain the asymptotic decay rate of the stationary tail distribution in an arbitrary direction. This result is an extension of the corresponding result for a certain two-dimensional reflecting random work without background processes, obtained by using the large deviation techniques and the singularity analysis methods. We also present a condition ensuring the sequence of the stationary probabilities geometrically decays without power terms, asymptotically. Asymptotic properties of the stationary tail distribution in the coordinate directions in a 2d-QBD process have already been studied in the literature. The results of this paper are also important complements to those results. [ABSTRACT FROM AUTHOR]

In this paper the impact of dual supply chain on a perishable inventory model with negative arrivals is evaluated. The perishable and replenishment rates of dual suppliers are distributed exponentially. Arrival process follows Poisson distribution and the probability for an ordinary customer is p and for the negative customer is q. Limiting distribution of the assumed model is obtained. Numerical results are presented for cost function and various system performance parameters. The impact of dual suppliers on the optimal reorder points will be useful in developing strategies for handling various perishable inventory problems with replenishment rates. [ABSTRACT FROM AUTHOR]

The queueing systems with a fixed batch size are commonly discussed in literatures. However, most real-life systems involve multiple types of servers with varying service capacities. To characterize this feature, random service capacity is considered in a double-ended queueing system in this paper, in which an arrival server can serve at most k customers with probability α k. Customers' threshold strategies are discussed based on the conditional expected waiting time and customers' beliefs in the number of servers in the orbit. Then, stationary analyses are conducted for this double-ended queueing system with random service capacity and balking customers, which includes establishing a sufficient stability condition, calculating the stationary distribution for system states and stationary performance measures. Finally, numerical examples are presented to explore how performance measures are influenced by system parameters. [ABSTRACT FROM AUTHOR]

In this paper we analyzed an inventory model with two-suppliers, finite life times, multiple working vacations and customers who arrive according to RCE process. Perishable and replenishment rates of two-suppliers are exponentially distributed. The server takes exponential working vacations when the queue is empty. Arrival process follows Poisson distribution and the probability for an ordinary customer is p and for negative customer is q. Limiting distribution of the assumed model is obtained. Numerical results are presented for cost function and various system performance parameters. The impact of two-suppliers on the optimal reorder points will be useful in developing strategies for handling various perishable inventory problems with replenishment rates. [ABSTRACT FROM AUTHOR]

This paper considers a single server queueing system with working breakdowns and delaying repair under a Bernoulli-schedule-controlled policy. At a breakdown instant, the system either goes to repair period immediately with probability p, or continues to provide auxiliary service for the current customers with probability q = 1 − p. While the system resides in the auxiliary service period, it may go to repair period if there is no customer at the epoch of service completion or the occurrence of breakdown. By using the matrix analytic method and the spectral expansion method, we respectively obtain the steady state distribution to make the straightforward computation of performance measures and the Laplace-Stieltjes transform of the stationary sojourn time of an arbitrary customer. In addition, some numerical examples are presented to show the impact of parameters on the performance measures. [ABSTRACT FROM AUTHOR]

We consider a discrete-time two-dimensional process {(X1,n,X2,n)} on Z+2 with a supplemental process {Jn} on a finite set, where the individual processes {X1,n} and {X2,n} are both skip-free. We assume that the joint process {Yn}={(X1,n,X2,n,Jn)} is Markovian and that the transition probabilities of the two-dimensional process {(X1,n,X2,n)} are modulated depending on the state of the supplemental process {Jn}. This modulation is space homogeneous except for the boundaries of Z+2. We call this process a discrete-time two-dimensional quasi-birth-and-death process. Under several conditions, we obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions. [ABSTRACT FROM AUTHOR]

We consider Markovian multiserver retrial queues where a blocked customer has two opportunities for abandonment: at the moment of blocking or at the departure epoch from the orbit. In this queueing system, the number of customers in the system (servers and buffer) and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose stationary distribution is expressed in terms of a sequence of rate matrices. Using a simple perturbation technique and a matrix analytic method, we derive Taylor series expansion for nonzero elements of the rate matrices with respect to the number of customers in the orbit. We also obtain explicit expressions for all the coefficients of the expansion. Furthermore, we derive tail asymptotic formulas for the joint stationary distribution of the number of customers in the system and that in the orbit. Numerical examples reveal that the tail probability of the model with two types of nonpersistent customers is greater than that of the corresponding model with one type of nonpersistent customers. They also reveal that the series expansion with a few terms can be used to obtain an accurate approximation to the stationary distribution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a continuous review perishable inventory system wherein demands arrive according to the Poisson process, each demanding exactly one unit of inventory item and the life time of each item is assumed to be exponential. The operating policy is ( s, S ) policy, i.e., whenever the inventory level drops to s , an order for Q (= S − s ) items is placed. The ordered items are received after a random time which is distributed as exponential. The service may be interrupted according to the Poisson process in which case it restarts after an exponentially distributed time. The demands that occur during the server breakdown period or stock-out period may turn out to be ordinary or a negative demand and then they enter into the orbit of infinite size. These orbiting demands send out a signal to compete for their demand which is distributed as exponential. The matrix analytic method is used for the steady state distribution of the model. Various performance measures and cost analysis are shown with numerical results. [ABSTRACT FROM AUTHOR]

The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network. The data packets (customers), belonging to different classes, arrive according to a batch marked the Markovian arrival process (BMMAP). The inventory considered is of multi-type (different types of channels becoming available) and are generated according to a marked Markovian arrival process (MMAP). The number of distinct types of inventory and that of the customers are the same. Arriving customers are allowed to wait in finite buffers of each category which are reserved for distinct classes of customers except for the most general class, which is provided with an infinite waiting space. The number of servers also equals the number of distinct types of inventory. When items of a particular type arrive in the inventory, the service starts, providing the buffer of customers of the corresponding class is non-empty. The service can be viewed as a selection process with Coxian distributed service times. The system is analyzed using the matrix analytic method and performance measures are obtained. The model is illustrated with suitable numerical examples. [ABSTRACT FROM AUTHOR]

We consider a discrete-time two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ on $\mathbb{Z}_{+}^{2}$ with a background process { J} on a finite set, where individual processes $\{L_{n}^{(1)}\}$ and $\{L_{n}^{(2)}\}$ are both skip free. We assume that the joint process $\{Y_{n}\}=\{(L_{n}^{(1)},L_{n}^{(2)},J_{n})\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ are modulated depending on the state of the background process { J}. This modulation is space homogeneous, but the transition probabilities in the inside of $\mathbb{Z}_{+}^{2}$ and those around the boundary faces may be different. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process, and obtain the decay rates of the stationary distribution in the coordinate directions. We also distinguish the case where the stationary distribution asymptotically decays in the exact geometric form, in the coordinate directions. [ABSTRACT FROM AUTHOR]

Recently, Tarabia (Appl. Math. Model., 2008, 802) studied the steady-state probabilities of two parallel queues with jockeying and restricted capacities, using the matrix-analytical technique. In this paper, the differential-difference equations which describe the transient state case are derived. Using the fourth order Runge-Kutta method and randomization methods, transient-state probabilities of the Tarabia (2008) model are computed. It is shown that these two methods are closely related, but that the randomization method is superior to the Runge-Kutta method. In the transient case, a numerical comparison between Tarabia's model and Conolly's (J. Appl. Prob., 1984, 394) model is presented to highlight the effect of jockeying on the average of the queue length and the waiting time. Finally, some illustrative numerical results are provided, and conclusions are presented. [ABSTRACT FROM AUTHOR]

Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated. [ABSTRACT FROM AUTHOR]

This paper presents two methods to calculate the response time distribution of impatient customers in a discrete-time queue with Markovian arrivals and phase-type services, in which the customers’ patience is generally distributed (i.e., the D-MAP/PH/1 queue). The first approach uses a GI/M/1 type Markov chain and may be regarded as a generalization of the procedure presented in Van Houdt [14] for the D-MAP/PH/1 queue, where every customer has the same amount of patience. The key construction in order to obtain the response time distribution is to set up a Markov chain based on the age of the customer being served, together with the state of the D-MAP process immediately after the arrival of this customer. As a by-product, we can also easily obtain the queue length distribution from the steady state of this Markov chain. We consider three different situations: (i) customers leave the system due to impatience regardless of whether they are being served or not, possibly wasting some service capacity, (ii) a customer is only allowed to enter the server if he is able to complete his service before reaching his critical age and (iii) customers become patient as soon as they are allowed to enter the server. In the second part of the paper, we reduce the GI/M/1 type Markov chain to a Quasi-Birth-Death (QBD) process. As a result, the time needed, in general, to calculate the response time distribution is reduced significantly, while only a relatively small amount of additional memory is needed in comparison with the GI/M/1 approach. We also include some numerical examples in which we apply the procedures being discussed. [ABSTRACT FROM AUTHOR]

We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n -th arrival–-depending on whether we are studying the transient queue length or waiting time distribution–-by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained. In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n -th customer. Other distributions, e.g., the amount of work left behind by the n -th customer, that can be acquired in a similar way, are briefly touched upon. Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs–-as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved–-offering new perspectives to the transient analysis of practical queueing systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion. [ABSTRACT FROM AUTHOR]

We study a state-dependent bulk matching queueing system with impatient customers, where customers and servers visit the system from both sides. Servers provide services in a batch with a maximum size and take matching customers away instantly. To characterize such a queueing system, the corresponding Markov process is constructed by the number of complete batches of customers and the number of remaining customers in the incomplete batch. By analyzing this system, we find it difficult to obtain the joint stationary distribution of the Markov process. Therefore, we consider the tail asymptotics for the joint probabilities. Using the matrix analytic method and censoring technique, we obtain the one-term and general expansions for the nonzero elements of the rate matrices, where the coefficients of the expansions are presented in the closed form. On the basis of these expansion formulae, the exact tail asymptotic result for the joint stationary probabilities is derived. [ABSTRACT FROM AUTHOR]