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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]

Among the waiting models proposed in the literature most of them focus on homogeneous arrival rate. This paper proposes a single server waiting line with modulated arrival rate of small and large size of traffic sources. Modulation of arrival rate induces the server to shift the service rate that follow phase type distribution. Using Neut’s versatile point process under quasi birth death environment we present the steady state analysis of the system. We provide examples numerically to perform the results and represent it in graphical form.

In this paper, we analyze an automated manufacturing system modeled as a queueing system with multiple servers, differentiated vacation (DV) policy, customer impatience, and server failures. We consider the waiting servers policy, where the servers wait for a random duration before leaving for a vacation once the system gets empty. Upon their return, if no customers are waiting, the servers may leave for a second type-2 vacation of shorter duration. The system may experience breakdowns and require immediate repair. During the repair period, customers are served at a lower rate, and they become impatient during busy, vacation, and repair periods, abandoning the queue if their timer expires. Further, based on the queue length, arriving customers may either join the system or balk with some probability. We use the matrix analytic method to derive the steady-state probability of the system and several performance measures. Moreover, we illustrate the effect of different system parameters on various performance measures. Additionally, we develop a cost model and optimize the system capacity, number of servers, and service rates such that cost per unit time is minimized. Finally, we analyze the impact of system parameters on the cost through numerical examples.

The service discipline of weighted fair queueing (WFQ) has been practically used in many telecommunication systems. The development of analytically tractable models with WFQ discipline to determine their performances is vitally important. Motivated by a practical project on routers' optimization from the Huawei Company, this paper deals with a two-class single server MAP/PH/1 queue with a finite capacity under the WFQ discipline, which is little investigated in the literature. To capture the most generality of stochastic factors in traffic traces, we assume that the arrive process is a Markovian arrival process (MAP) and the service time obeys a phase-type (PH) distribution. This paper focuses on the performance analysis of our proposed queueing system and its application to traffic flow performances. We model this system as a five-dimensional Markov chain and use the matrix-geometric method to derive its stationary probabilities and performance measures, such as the average queue length and sojourn time of each class of customers. In order to validate our analysis results, we further develop a simulator to evaluate the system performance with a real data of Internet packet traces. Experiment comparison demonstrates the effectiveness and applicability of our theoretical model. [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]

This paper deals with a M/M/c queueing system with waiting servers, balking, reneging, and K-variant working vacations subjected to Bernoulli schedule vacation interruption. Whenever the system is emptied, the servers wait for a while before synchronously going on vacation during which services are offered with a lower rate. We obtain the steady-state probabilities of the system using the matrix-geometric method. In addition, we derive important performance measures of the queueing model. Moreover, we construct a cost model and apply a direct search method to get the optimum service rates during both working vacation and regular working periods at lowest cost. Finally, numerical results are provided. [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]

This paper studies a discrete-time single server batch arrival and batch service queueing model with unlimited waiting space. The discrete-time batch Markovian arrival process and discrete-time batch Markovian service process, respectively, manage the arrival and service processes. We adopt the UL-type RG-factorization approach based on censoring methodology for variable size batch service queue to calculate the stationary probability vectors of the transition probability matrix with general structure Markov chain at outside observer’s epoch. We reblock the transition probability matrix to its desired M/G/1 structure to find the stationary probability vectors at outside observer’s epoch for fixed size batch service queue using the matrix analytic method. We also develop relationships to determine probability vector expressions for other important time epochs such as pre-arrival, intermediate, post-departure, and random epochs. The most challenging aspect of our study is to obtain the probability mass functions of sojourn time in the system for both the variable and fixed size batch service queues. We use our suggested queueing model to derive the results of several specific well-known queueing models. We also discuss about possible managerial implication of our model to produce fruit juices in manufacturing industry. We present computational experience based on the execution of parametrized experiments with various categories in order to validate the analytical results reported in this study. [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]

Among the waiting models proposed in the literature most of them focus on homogeneous arrival rate. This paper proposes a single server waiting line with modulated arrival rate of small and large size of traffic sources. Modulation of arrival rate induces the server to shift the service rate that follow phase type distribution. Using Neut's versatile point process under quasi birth death environment we present the steady state analysis of the system. We provide examples numerically to perform the results and represent it in graphical form. [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.

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 Cj 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 L1 where L1

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 perishable inventory system in which customers arrive in a homogeneous Poisson stream. The system has a production unit which produces a single item in an exponentially distributed time interval. At the time of arrival, a customer leads to service if the server is available with a positive level of inventory. Otherwise, the customer goes to a waiting place(orbit) of infinite capacity with pre-determined probability or exits the system with complementary probability. Each customer in the orbit tries to access the server in an exponentially distributed time interval. After every unsuccessful retrial, the customer returns to the orbit with a pre-allotted probability or is lost forever with complementary probability. An algorithmic solution to the problem is obtained using Matrix Analytic Method. The mean number of customer loss before and after entering the system, the rate of successful retrials among overall retrials and some other performance measures of the system are derived. The impacts of system parameters on different measures are numerically studied. A suitable cost function is constructed and the optimum control policy is numerically obtained. [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.

This paper investigates an optimal K-policy for a two-server Markovian queueing system M / (M 1 , M 2) / 2 / (B 1 , B 2) , with one fast server S 1 and one slow server S 2 , using the matrix analytic method. Two buffers B 1 and B 2 are organized to form waiting lines of customers in which, buffer B 1 is of finite size K (< ∞) and buffer B 2 is of infinite capacity. Buffer B 1 stalls customers who arrive when the system size (queue + service) is less than (K + 1) and dispatches a customer to the fast server S 1 only after S 1 completes its previous service. This K-policy is of threshold type which deals with controlling of informed customers and hence the customers have better choice of choosing the fast server routing through the buffer B 1 . The (K + 2) -nd customer who arrives when the number of customers present in the system is exactly (K + 1) has the Hobson's choice of getting service from the slow server S 2. Buffer B 2 accommodates other customers who arrive when the number of customers present in the system is (K + 2) or more and feeds them one after another to either buffer B 1 or the sever S 2 whichever event can first accept the customer at the head-of-the-line in B 2 . Queue length processes of interest are (1) q 1 = lim t → ∞ X 1 (t) and (2) q 2 = lim t → ∞ X 2 (t) , where X 1 (t) represents the number of customers who are in the buffers B 1 and B 2 and also in the service with server S 1 at time 't' and X 2 (t) represents the number of customers available with server S 2 only. The bi-variate random sequence X (t) = (X 1 (t) , X 2 (t)) of the system size (queue + service) forms a quasi-birth and death process (QBD). Steady state characteristics, and some of the performance measures such as the expected queue length, the probability that each server is busy etc are obtained. Numerical illustrations are provided based on the average cost function to explore the methodology of finding the best K-policy which minimizes the mean sojourn time of customers. [ABSTRACT FROM AUTHOR]

This paper analyses two (s, S) production inventory systems with different service rates. The time for producing and adding each item to the inventory is exponentially distributed. Arrivals of customers follow a Poisson process and service times are exponentially distributed. Upon arrival, the customers enter into a buffer of finite capacity. A potential customer who finds the buffer full, moves to an orbit. They can retry from there and inter-retrial times are exponentially distributed. When the inventory level depletes to s, service is given at a reduced rate. This is to minimise customers' loss from the system. Various system performance measures are computed using matrix analytic method. The effect of the underlying parameters on the performance measures are also analyzed. The minimum expected total cost by varying different parameters is computed numerically. The optimum values of s and S of the efficient model are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a limited clearing queueing model with an orbit and non-persistent customers, in which the space of service station is finite and the server can serve all customers in the service station simultaneously. If a customer (new arrival or retrial) finds the station is full, he/she will decide whether join the orbit or not with respective probabilities. We first analyze the necessary and sufficient condition for the stability of this system, then using matrix geometric analytic method and spectral expansion method, we obtain the stationary distribution of this model. Besides, we also give many important performance measures of this system which help managers to make wisdom managerial decisions and designs, such as the mean size of service station, the mean queue length of the orbit and the mean sojourn time of an arbitrary customer. Finally, some numerical examples are provided to show the impacts of various parameters on the system performance measures. [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]

This study considers a single-server Markovian working vacation queuing system with Bernoulli vacation interruptions. Based on a linear reward-cost structure, the customer strategic joining behavior is analyzed under different information levels available to the arriving customers, namely fully observable, almost unobservable, and fully unobservable. For these cases, we first obtain the system stationary distribution. Thereafter, we determine the customer equilibrium strategies and compare them numerically with socially optimal strategies. [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]

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.

In this paper, we consider a single-server multi-queue polling system with unlimited-size batch service (so called ‘Israeli queue’) operating in a multi-phase random environment. The polling system consists of a service region and a waiting region, and the external environment evolves through time, i.e., when the external environment is in state i, after a period time, it stays in this state or makes a transition from this state to its adjacent ones. By using matrix analytic method and spectral expansion method, stationary probabilities are derived for computations of performance measures and the conditional waiting times of customers in waiting region. In addition, some numerical examples are presented to show the impact of parameters on performance measures. [ABSTRACT FROM AUTHOR]

This paper analyses an inventory model with positive service time and retrial of customers. The customers arrive according to a Markovian Arrival Process with representation (D0,D1). The service times are assumed to be of Phase type distribution with representation η,U. When the inventory level depletes to the reorder point s, the service is given at a reduced rate and the service time distribution has the representation η,αU, where 0<α<1. An arriving customer, who finds the inventory level zero or the server busy, enters into an orbit of infinite capacity and will retry for service from there. The lead time follows an exponential distribution with rate β. We analyze the system using Matrix Analytic Method. Some important performance measures in the steady state are obtained. A suitable cost function for the expected total cost is constructed and analyzed numerically and graphically. [ABSTRACT FROM AUTHOR]

This paper considers the comparison of two $$\left( {s,S} \right)$$ production inventory systems with retrials of unsatisfied customers. The time for producing and adding each item to the inventory is exponentially distributed with rate $$\beta$$ . However, a production rate $$\alpha \beta$$ higher than $$\beta$$ is used at the beginning of the production. The higher production rate will reduce customers' loss when inventory level approaches zero. The demand from customers is according to a Poisson process. Service times are exponentially distributed. Upon arrival, the customers enter into a buffer of finite capacity. An arriving customer, who finds the buffer full, moves to an orbit. They can retry from there and inter-retrial times are exponentially distributed. The two models differ in the capacity of the buffer. The aim is to find the minimum value of total cost by varying different parameters and compare the efficiency of the models. The optimum value of $$\alpha$$ corresponding to minimum total cost is an important evaluation. Matrix analytic method is used to find an algorithmic solution to the problem. We also provide several numerical or graphical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the optimal control of a retrial queueing system with finite buffer K. At any decision epoch, if the buffer is full, the controller have to make two decisions: one is for the new arrivals, to decide whether they are allowed to join the orbit or not (admission control); the other one is for the repeated customers, to decide whether they are allowed to get back to the orbit or not (retrial control). The problems are constructed as a Markov decision process. We show that the optimal policy has a threshold-type structure and the thresholds are monotonic in operating parameters and various cost parameters. Furthermore, based on the structure of the optimal policy, we construct a performance evaluation model for computing efficiently the thresholds. The expression of the expected cost is given by solving the quasibirth- and-death (QBD) process. Finally, we provide some numerical results to illustrate the impact of different parameters on the optimal policy and average cost. [ABSTRACT FROM AUTHOR]

Crowdsourcing is getting popular after a number of industries such as food, consumer products, hotels, electronics, and other large retailers bought into this idea of serving customers. In this paper, we introduce a multi-server queueing model in the context of crowdsourcing. We assume that two types, say, Type 1 and Type 2, of customers arrive to a c-server queueing system. A Type 1 customer has to receive service by one of c servers while a Type 2 customer may be served by a Type 1 customer who is available to act as a server soon after getting a service or by one of c servers. We assume that a Type 1 customer will be available for serving a Type 2 customer (provided there is at least one Type 2 customer waiting in the queue at the time of the service completion of that Type 1 customer) with probability $$p, 0 \le p \le 1$$ . With probability $$q = 1 - p$$ , a Type 1 customer will opt out of serving a Type 2 customer provided there is at least one Type 2 customer waiting in the system. Upon completion of a service a free server will offer service to a Type 1 customer on an FCFS basis; however, if there are no Type 1 customers waiting in the system, the server will serve a Type 2 customer if there is one present in the queue. If a Type 1 customer decides to serve a Type 2 customer, for our analysis purposes that Type 2 customer will be removed from the system as Type 1 customer will leave the system with that Type 2 customer. Under the assumption of exponential services for both types of customers we study the model in steady state using matrix analytic methods and establish some results including explicit ones for the waiting time distributions. Some illustrative numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with a stochastic model for the spread of an epidemic in a closed homogeneously mixing population when an infective can go through several stages of infection before being removed. The transitions between stages are governed by either a Markov process or a semi-Markov process. An infective of any stage makes contacts amongst the population at the points of a Poisson process. Our main purpose is to derive the distribution of the final epidemic size and severity, as well as an approximation by branching, using simple matrix analytic methods. Some illustrations are given, including a model with treatment discussed by Gani (2006). [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]

In this paper, we consider a single server queueing system wherein demands arrive according to Poisson process, each demanding exactly one unit of inventory item. The operating policy is ( s, S) policy in which an order is placed for a quantity up to S whenever the inventory level falls to s. The ordered items are received after a random time which is exponentially distributed. The service may be interrupted according to 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 negative and they enter into the orbit of infinite size. These demands retry after a random time, which is again exponentially distributed. Matrix geometric solution is obtained 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]

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]