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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 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]

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]

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]

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 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]

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]

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]

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]