Multi-threshold Control of the BMAP/SM/1/K Queue with Group Services

We consider a finite capacity queue in which arrivals occur according to a batch Markovian arrival process (BMAP). The customers are served in groups of varying sizes. The services are governed by a controlled semi-Markovian process according to a multithreshold strategy. We perform the steady-state analysis of this model by computing (a) the queue length distributions at departure and arbitrary epochs, (b) the Laplace-Stieltjes transform of the sojourn time distribution of an admitted customer, and (c) some selected system performance measures. An optimization problem of interest is presented and some numerical examples are illustrated

Pseudo steady-state period in non-stationary infinite-server queue with state dependent arrival intensity

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation

Queueing System with Potential for Recruiting Secondary Servers

In this paper, we consider a single server queueing system in which the arrivals occur according to a Markovian arrival process (MAP). The served customers may be recruited (or opted from those customers’ point of view) to act as secondary servers to provide services to the waiting customers. Such customers who are recruited to be servers are referred to as secondary servers. The service times of the main as well as that of the secondary servers are assumed to be exponentially distributed possibly with different parameters. Assuming that at most there can only be one secondary server at any given time and that the secondary server will leave after serving its assigned group of customers, the model is studied as a QBD-type queue. However, one can also study this model as a G I/M/1-type queue. The model is analyzed in steady state, and a few illustrative numerical examples are presented

Optimal control for a BMAP/G/1 queue with two service modes

Queueing models with controllable service rate play an important role in telecommunication systems. This paper deals with a single-server model with a batch Markovian arrival process (BMAP) and two service modes, where switch-over times are involved when changing the service mode. The embedded stationary queue length distribution and the explicit dependence of operation criteria on switch-over levels and derived

Eastern Europe’s “Transitional Industry”? : Deconstructing the Early Streletskian

Acknowledgements We are very grateful to many friends and colleagues for discussions and various help, including Yuri Demindenko, Evgeny Giria, Brad Gravina, Anton Lada, Sergei Lisitsyn and Alexander Otcherednoy. Needless to say, they may or may not agree with our conclusions. We are also thankful to Jesse Davies and Craig Williams for the help with the illustrations and figures. Ekaterina Petrova kindly helped with ID’ing some of the sampled bones. We thank the staff of the Oxford Radiocarbon Accelerator Unit at the University of Oxford for their support with the chemical preparation and the measurement of the samples. We are also grateful to the three anonymous reviewers for their thoughtful and constructive comments, which helped improve the paper. This paper is a contribution to Leverhulme Trust project RPG-2012-800. The research leading to some of our radiocarbon results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013); ERC grant 324139 “PalaeoChron” awarded to Professor Tom Higham. AB and AS acknowledge Russian Science Foundation grant number 20-78-10151 and Russian Foundation of Basic Research grant numbers 18-39-20009 and 20-09-00233 for support of their work. We also acknowledge the participation of IHMC RAS (state assignment 0184-2019-0001) and ZIN RAS (state assignment АААА-А19-119032590102-7).Peer reviewedPublisher PD

Multi-dimensional quasitoeplitz Markov chains

This paper deals with multi-dimensional quasitoeplitz Markov chains. We establish a sufficient equilibrium condition and derive a functional matrix equation for the corresponding vector-generating function, whose solution is given algorithmically. The results are demonstrated in the form of examples and applications in queues with BMAP-input, which operate in synchronous random environment

Optimal Hysteretic Control for the BMAP/G/1 System with Single and Group Service Modes

In this paper, we consider a single server queuing model with an infinite buffer in which customers arrive according to a batch Markovian arrival process (BMAP). The services are offered in two modes. In mode 1, the customers are served one at a time and in mode 2 customers are served in groups of varying sizes. Various costs for holding, service and switching are imposed. For a given hysteretic strategy, we derive an expression for the cost function from which an optimal hysteretic control can be obtained. Illustrative numerical examples are presented

Analysis of Multiserver Retrial Queueing System with Varying Capacity and Parameters

A multiserver queueing system, the dynamics of which depends on the state of some external continuous-time Markov chain (random environment, RE), is considered. Change of the state of the RE may cause variation of the parameters of the arrival process, the service process, the number of available servers, and the available buffer capacity, as well as the behavior of customers. Evolution of the system states is described by the multidimensional continuous-time Markov chain. The generator of this Markov chain is derived. The ergodicity condition is presented. Expressions for the key performance measures are given. Numerical results illustrating the behavior of the system and showing possibility of formulation and solution of optimization problems are provided. The importance of the account of correlation in the arrival processes is numerically illustrated

A Batch Markovian Queue with a Variable Number of Servers and Group Services

In this paper, we consider a multi-server queuing model with a finite buffer in which customers arrive according to a batch Markovian arrival process (BMAP). These customers are served in groups of varying sizes ranging from a predetermined value L through a maximum size, K. The service times are exponentially distributed. The number of servers in the system at any given time varies between a lower limit and an upper limit. The steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. Some interesting numerical examples are discussed

A Multi-Server Retrial Queue with BMAP Arrivals and Group Services

In this paper, we consider a c-server queuing model in which customers arrive according to a batch Markovian arrival process (BMAP). These customers are served in groups of varying sizes ranging from a predetermined value L through a maximum size, K. The service times are exponentially distributed. Any customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed with parameter θ. Under a full access policy freed servers offer services to orbiting customers in groups of varying sizes. This multi-server retrial queue under the full access policy is a QBD process and the steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. Some interesting numerical examples are discussed