Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers

Abstract

This study investigates an infinite capacity Markovian queue with a single unreliable service station, in which the customers may balk (do not enter) and renege (leave the queue after entering). The unreliable service station can be working breakdowns even if no customers are in the system. The matrix-analytic method is used to compute the steady-state probabilities for the number of customers, rate matrix and stability condition in the system. The single-objective model for cost and bi-objective model for cost and expected waiting time are derived in the system to fit in with practical applications. The particle swarm optimisation algorithm is implemented to find the optimal combinations of parameters in the pursuit of minimum cost. Two different approaches are used to identify the Pareto optimal set and compared: the epsilon-constraint method and nondominate sorting genetic algorithm. Compared results allow using the traditional optimisation approach epsilon-constraint method, which is computationally faster and permits a direct sensitivity analysis of the solution under constraint or parameter perturbation. The Pareto front and non-dominated solutions set are obtained and illustrated. The decision makers can use these to improve their decision-making quality.

Timely Reliability Analysis of Virtual Machines Considering Migration and Recovery in an Edge Server

“…However, the failure of a server or a VM can reduce the processing capability of the edge node directly and increase latency [ 12 , 40 , 41 ]. To model the failure and repair of VMs, VMs can be treated as unreliable and repairable queue servers based on the multi-server queueing models with unreliable and repairable servers [ 20 , 21 , 22 , 23 , 26 ]. For instance, Ke et al [ 26 ] and Chakravarthy et al [ 23 ] analyzed multi-server queueing models considering unreliable servers and vacation policy.…”

“…All this considered, present studies on VM delay did not investigate the influence of VM migration on the total delay at the VMs, for the queueing models of VMs ignored the failure and migration of VMs [ 24 , 25 , 27 ], and the VM migration delay was measured separately from the total delay [ 12 , 13 , 14 , 42 , 44 , 45 , 46 , 47 , 53 ]. Meanwhile, due to the unique process of failure, migration, and recovery of VMs, present queueing models with unreliable servers [ 20 , 21 , 22 , 23 , 26 ] cannot be directly applied to analyze the VM delay. To analyze the timely reliability of the VMs and the network, we build a queueing model considering the whole process of failure, migration, and recovery of VMs and propose a method to obtain the PDF of the delay at the edge server.…”

“…To measure the timely reliability of edge computing more precisely, we need to build a queueing model considering failure, migration, and reboot of VM. Present mathematical models [ 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 ] cannot take the migration of VM into account and neglect the migration downtime. Therefore, to analyze the edge computing system considering migration and recovery of VM, the essential migration-related states of VM need to be defined in a new evaluation model.…”

Analysis of a renewal batch arrival queue with a fault-tolerant server using shift operator method

“…Liu and Song ( 2014 ) extended the idea in Kalidass and Kasturi ( 2012 ) to the Markovian batch arrival queue. Applying the matrix-geometric method (Neuts 1981 ), Liou ( 2015 ) also investigated M/M/1 unreliable queue subject to working breakdowns and impatient customers. Other recent works on queues with working breakdowns can be found in Li et al.…”

An M/G/1 Retrial G-Queue with General Retrial Times and Working Breakdowns

“…Using the matrix geometric method, Ma et al [16] computed the steady-state distribution of an M/M/1 queue with multiple vacations and working breakdowns. For more queueing models with working breakdowns, readers can also refer to Li et al [17], Liou [18], Chen et al [19] and Yen et al [20]. Note that the concept of working breakdown is different from the concept of working vacation introduced by Servi and Finn [21].…”