We consider a two-queue polling model in which customers upon arrival join the shorter of two queues. Customers arrive according to a Poisson process and the service times in both queues are independent and identically distributed random variables having the exponential distribution. The two-dimensional process of the numbers of customers at the queue where the server is and at the other queue is a two-dimensional Markov process. We derive its equilibrium distribution using two methodologies: the compensation approach and a reduction to a boundary value problem.
In this paper we present a detailed analysis of queueing models with vacations and impatient customers, where the source of impatience is the absence of the server. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers abandon the system simultaneously. This is, for example, the case in remote systems where customers may decide to abandon the system, when a transport facility becomes available.
We consider a system in which customers join upon arrival the shortest of two single-server queues. The interarrival times between customers are Erlang distributed and the service times of both servers are exponentially distributed. Under these assumptions, this system gives rise to a Markov chain on a multi-layered quarter plane. For this Markov chain we derive the equilibrium distribution using the compensation approach. The obtained expression for the equilibrium distribution matches and refines heavy-traffic approximations and tail asymptotics obtained earlier in the literature.
In this article we present a sequence of random variables with weighted tail distribution functions, constructed based on the relevation transform. For this sequence, we prove several recursive formulas and connections to the residual entropy through the unifying framework of the Dickson–Hipp operator. We also give some numerical examples to evaluate our results.
We consider a queueing system consisting of two nonidentical exponential servers, where each server has its own dedicated queue and serves the customers in that queue FCFS. Customers arrive according to a Poisson process and join the queue promising the shortest expected delay, which is a natural and near-optimal policy for systems with nonidentical servers. This system can be modeled as an inhomogeneous random walk in the quadrant. By stretching the boundaries of the compensation approach we prove that the equilibrium distribution of this random walk can be expressed as a series of product forms that can be determined recursively. The resulting series expression is directly amenable to numerical calculations and it also provides insight into the asymptotic behavior of the equilibrium probabilities as one of the state coordinates tends to infinity.
In this paper, we study birth/immigration-death processes under mild (binomial) catastrophes. We obtain explicit expressions for both the time-dependent (transient) and the limiting (equilibrium) factorial moments, which are then used to construct the transient and equilibrium distribution of the population size. We demonstrate that our approach is also applicable to multidimensional systems such as stochastic processes operating under a random environment and other variations of the model at hand. We also obtain various stochastic order results for the number of individuals with respect to the system parameters, as well as the relaxation time.
In this paper we present a detailed analysis of a single server Markovian queue with impatient customers. Instead of the standard assumption that customers perform independent abandonments, we consider situations where customers abandon the system simultaneously. Moreover, we distinguish two abandonment scenarios; in the first one all present customers become impatient and perform synchronized abandonments, while in the second scenario we exclude the customer in service from the abandonment procedure. Furthermore, we extend our analysis to the M/M/c queue under the second abandonment scenario.For these models we carry out an extensive analysis including the stationary, the busy period and the conditional sojourn time distributions deriving exact formulas and iterative algorithmic schemes. We also obtain explicit results under various limiting regimes that demonstrate the effect of the level of synchronization on the performance of the systems.
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