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A single server queueing system with working vacation operating in a multi-level environment controlled by a random switch is considered. The environment has N levels and the random switch puts the server in one of the N levels to perform a service. The server resides in a level and reports to the random switch immediately after completing the service of the last customer in the queue. The random switch initiates an assignment process immediately at the arrival of a customer. The assignment process takes a random time and immediately at the end of the assignment interval, the server is put to operate in any one of the N levels with a positive probability. In each level r of the environment, the queueing system behaves like an M(λr,1)/M(μr,1)/1 queue and switches to a working vacation at a random time and serves with a lesser rate μr,2. During the working vacation, all newly arriving customers are lost. When the server is with the control switch, customers are permitted to join in the queue. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made. [ABSTRACT FROM AUTHOR]

We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of 2 + ϵ moments. [ABSTRACT FROM AUTHOR]

The performance analysis of the classical M / G / 1 queue, under a general mixed joining/balking strategy was carried out recently by Kerner (Stoch Mod 24:364–375, 2008), who used an analytic approach based on the supplementary variable method. The tractability of the corresponding queueing system with state-dependent arrival rates is particularly significant, as it has important applications in situations where the customers are strategic. In this paper, we present an alternative path for the analysis of the same system, using purely probabilistic arguments. [ABSTRACT FROM AUTHOR]

We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process. [ABSTRACT FROM AUTHOR]

The stochastic runoff-runon process models the volume of infiltration excess runoff from a hillslope via the overland flow path. Spatial variability is represented in the model by the spatial distribution of rainfall and infiltration, and their “correlation scale”, that is, the scale at which the spatial correlation of rainfall and infiltration become negligible. Notably, the process can produce runoff even when the mean rainfall rate is less than the mean infiltration rate, and it displays a gradual increase in net runoff as the rainfall rate increases. In this paper we present a number of contributions to the analysis of the stochastic runoff-runon process. Firstly we illustrate the suitability of the process by fitting it to experimental data. Next we extend previous asymptotic analyses to include the cases where the mean rainfall rate equals or exceeds the mean infiltration rate, and then use Monte Carlo simulation to explore the range of parameters for which the asymptotic limit gives a good approximation on finite hillslopes. Finally we use this to obtain an equation for the mean net runoff, consistent with our asymptotic results but providing an excellent approximation for finite hillslopes. Our function uses a single parameter to capture spatial variability, and varying this parameter gives us a family of curves which interpolate between known upper and lower bounds for the mean net runoff. [ABSTRACT FROM AUTHOR]

In operations research and networking problems, random-order-of-service (ROS) provides a well-known alternative to the classic first-come-first-served (FCFS) policy. Models with ROS policy are typically harder to analyze than their FCFS counterparts, with some performance measures notoriously hard to obtain. While significant progress has been realized in the analysis of random-order-of-service models with homogeneous customer service demands, the impact of heterogeneous customer demand is largely unknown. The current contribution studies this impact, with a discrete-time random-order-of-service queue serving customers with heterogeneous demands. Customer service times are independent random variables with type-dependent distribution. The numbers of new arrivals in each slot are independent and identically distributed over time, but can be type-correlated within a single slot. This corresponds to bursty input traffic generated by a finite number of sources, with one source for each type. The burstiness consists in correlation among sources (or types): the rate at which a source generates customers depends on the instantaneous rate of all other sources (and vice versa). Using a transform-based approach yields closed-form formulas for the first few moments of the customer waiting time. Facilitator is a multi-stage description of the customer sojourn, which allows establishing a relation between the so-called conditional waiting time and the actual steady-state customer waiting time. A somewhat unexpected result shows that, under certain conditions on the arrival and service processes, the random-order-of-service policy outperforms the first-come-first-served policy in terms of mean waiting time. A number of numerical examples illustrate this finding. [ABSTRACT FROM AUTHOR]

We study a batch arrival single server queueing system, where the server (service channel) provides one by one general service to customers. It is assumed that the service channel goes through random failures from time to time. As the result of a breakdown, the service of a customer in service is suspended, the service channel waits for the repairs to start and this waiting time termed as 'delay time' is assumed to be general. Further, the repair process involves two phases of repairs with different general repair time distributions. We derive the queue size distribution as well as mean number of customers in the system at a random epoch under the steady state conditions. In addition, we discuss some particular cases and derive some known results known earlier. [ABSTRACT FROM AUTHOR]

Loynes' distribution, which characterizes the one dimensional marginal of the stationary solution to Lindley's recursion, possesses an ultimately exponential tail for a large class of increment processes. If one can observe increments but does not know their probabilistic properties, what are the statistical limits of estimating the tail exponent of Loynes' distribution? We conjecture that in broad generality a consistent sequence of non-parametric estimators can be constructed that satisfies a large deviation principle. We present rigorous support for this conjecture under restrictive assumptions and simulation evidence indicating why we believe it to be true in greater generality. [ABSTRACT FROM AUTHOR]

A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ξ. If the system is not repaired before the customer’s timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions. [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]

We study a single server queueing system with general service time distribution and memoryless inter-arrival times. The arrival rate is not constant but varies with the number of customers in the system. A recursive formula for the conditional distribution of the remaining service time given the queue length is derived for an arbitrary epoch. It is also shown that this conditional distribution holds for arrival epochs. The recursion for the corresponding Laplace-Stieltjes transforms and expected values is given in a simple formula. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present two parallel queues with jockeying and restricted capacities. Each exponential server has its own queue, and jockeying among the queues is permitted. The capacity of each queue is restricted to L including the one being served. Customers arrive according to a Poisson process and on arrival; they join the shortest feasible queue. Moreover, if one queue is empty and in the other queue, more than one customer is waiting, then the customer who has to receive after the customer being served in that queue is transferred to the empty queue. This will prevent one server from being idle while the customers are waiting in the other queue. Using the matrix-analytical technique, we derive formulas in matrix form for the steady-state probabilities and formulas for other performance measures. Finally, we compare our new model with some of Markovian queueing systems such as Conolly’s model [B.W. Conolly, The autostrada queueing problems, J. Appl. Prob. 21 (1984) 394–403], queue and two of independent queues for the steady state solution. [Copyright &y& Elsevier]