Dmitry Efrosinin

Queueing system with a constant retrial rate, non-reliable server and threshold-based recovery

In this paper, we examine a Markovian queueing system which is composed of a retrial queue with constant retrial rate and a non-reliable server. Upon arrival a customer occupies the server if it is idle, otherwise it goes to the retrial queue. The customer at the head of the retrial queue is allowed to retry for service. When the server is busy, the server is subject to breakdowns or failures that occur according to a Poisson process. In the failed state the server can be repaired at a repair facility with exponential repair time with respect to the threshold policy: the repair starts when the number of customers in the system reaches some prespecified threshold level q [greater-or-equal, slanted] 1. We perform a steady-state analysis of the corresponding continuous-time Markov chain, derive mean performance characteristics and waiting time distribution as well as calculate optimal threshold level to minimize the long-run average losses for the given cost structure.

Performance Analysis of a Two-Server Heterogeneous Retrial Queue with Threshold Policy

Abstract In the paper we deal with a Markovian queueing system with two heterogeneous servers and constant retrial rate. The system operates under a threshold policy which prescribes the activation of the faster server whenever it is idle and a customer tries to occupy it. The slower server can be activated only when the number of waiting customers exceeds a threshold level. The dynamic behaviour of the system is described by a two-dimensional Markov process that can be seen as a quasi-birth-and-death process with infinitesimal matrix depending on the threshold. Using a matrix-geometric approach we perform a stationary analysis of the system and derive expressions for the Laplace transforms of the waiting time as well as arbitrary moments. Illustrative numerical results are presented for the threshold policy that minimizes the mean number of customers in the system and are compared with other heuristic control policies.

Sensitivity Analysis of Reliability Characteristics to the Shape of the Life and Repair Time Distributions

The present paper deals with some simple finite population queueing systems which are used to describe the cold redundancy systems. The systems are assumed to be in a complete failure state as soon as all of the units are failed. For such models the time dependent state probabilities and reliability function are analyzed. It is shown that the reliability function have a weak sensitivity to the shape of the life and repair time distributions and this sensitivity vanishes upon the probability of the complete failure state decreases.

Queueing model of a hybrid channel with faster link subject to partial and complete failures

This paper presents a Markovian queueing model for a hybrid channel consisting of two links with different throughputs. The busy faster link is assumed to be unreliable, with possible partial and complete failures. Partial failures lead to a reduction in the service rate, while complete failure stops the service. Repairs return the faster server to a non-failed state. The problem of the optimal allocation of customers between the servers is considered. The optimality of a threshold-based policy that depends on the failure state of the faster server is proved. The dynamic behaviour of the system for the given threshold policy is described by a four-dimensional Markov process that can be treated as a QBD process with a large number of boundary states. Stationary analysis of the system is performed by means of a matrix-geometric approach, and the main performance measures are derived.

On performance characteristics for queueing systems with heterogeneous servers

The problem of optimal control for queueing system (QS) with heterogeneous servers was considering by many authors. In [8] it was shown that the optimal with respect to the number of jobs in the system minimization policy is a threshold type one and it obliges to use the fastest free server if necessary. However, calculation of performance characteristics under optimal policy and analysis of its preference before some other policies rest out of the investigators’ interests. The purpose of this paper is to analyze a multi-server heterogeneous exponential queue. We demonstrate the methods for the calculation of the steady-state probabilities and deriving the waiting and sojourn time distributions. Some performance characteristics of such a system under the optimal control policy are calculated and compared with the same characteristics for the model under other heuristic control policies, e.g., the usage of the Fastest Free Server (FFS) or Random Server Selection (RSS).

Optimal control of M/M/1 queueing system with constant retrial rate and non-reliable removable server

The paper is concerned with the optimal control with respect to N-policy of the M/M/1 queue with constant retrial rate and non-reliable removable server. According to the N-policy, the server can start service only when the number of customers in the system reaches level N (N ≥ 1). We perform a steady-state analysis of the corresponding continuous-time Markov chain and calculate optimal threshold level to minimize the longrun average losses given cost structure.

Performance Analysis and Monotone Control of a Tandem Queueing System

A controllable tandem queueing system consists of two nodes in tandem of the type \(M/M/n_i\) and a controller. Customers arrive to the controller, who allocates them between the nodes. After service completion at node 2 the controller can allocate the customer waiting at node 1 to node 2. With probability \(p\) after a service completion at node 1 a failure occurs. In this case the customer from node 1 joins node 2. With complement probability \(1-p\) the service completion at node 1 is successful. For the given cost structure we formulate an optimal allocation problem to minimize the long-run average cost per unit of time. Using dynamic-programming approach we show the existence of thresholds which divides the state-space into two contiguous regions where the optimal decision is to allocate the customers to node 1 or to node 2. Some monotonicity properties of the dynamic-programming value function are established.

Change Point Detection in Piecewise Stationary Time Series for Farm Animal Behavior Analysis

Detection of abrupt changes in time series data structure is very useful in modeling and prediction in many application areas, where time series pattern recognition must be implemented. Despite of the wide amount of research in this area, the proposed methods require usually a long execution time and do not provide the possibility to estimate the real changes in variance and autocorrelation at certain points. Hence they cannot be efficiently applied to the large time series where only the change points with constraints must be detected. In the framework of the present paper we provide heuristic methods based on the moving variance ratio and moving median difference for identification of change points. The methods were applied for behavior analysis of farm animals using the data sets of accelerations obtained by means of the radio frequency identification (RFID).

Busy Period Analysis of a Queueing System with Breakdowns and Its Application to Wireless Network Under Licensed Shared Access Regime

Licensed shared access (LSA) framework is becoming one of the promising trends for future 5G wireless networks. Two main parties are involved in the process of sharing the frequency band – the primarily user (owner) and the secondary user (licensee). From the LSA licensee’s perspective, who has access to the band when the owner does not need it, the band is unreliable and its customers (e.g. users of wireless network) suffer from possible service interruptions. This can only occur when there is at least one customer in service (i.e. in busy period). The aim of this paper is to estimate the impact of the LSA band unreliability to the LSA licensee within the period when some interruptions are possible. The metric is the relation between the number of service interruptions and the number of customers served during a busy period. We model the occupancy of the LSA band as a multi-server homogeneous queueing system with finite and infinite buffer size and deal with the busy period analysis. The system is analyzed in steady state by deriving expressions in terms of the Laplace-Stiltjes transforms for the continuous time distributions and in terms of the probability generating functions for the number of customers served during a busy period and number of failures which occur in a busy period. The non-monotonic nature of some probabilistic measures is identified. Some illustrative examples are added to the paper.

Optimal Control of a Two-Server Heterogeneous Queueing System with Breakdowns and Constant Retrials

Heterogeneous servers which can differ in service speed and reliability are becoming more popular in the modelling of modern communication systems. For a two-server queueing system with one non-reliable server and constant retrial discipline we formulate an optimal allocation problem for minimizing a long-run average cost per unit of time. Using a Markov decision process formulation we prove a number of monotone properties for the increments of the dynamic-programming value function. Such properties imply the optimality of a two-level threshold control policy. This policy prescribes the usage of a less productive server if the number of customers in the queue becomes higher than a predefined level which depends on the state of a non-reliable more powerful server. We provide also a heuristic solution for the optimal threshold levels in explicit form as a function of system parameters.