Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

Abstract

This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phase i with probability qi, i = 1, 2, …, d. Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.

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 Vacation Fluid M/M/1 Queue in Multi-Phase Random Environment

“…All these results were obtained for the queue, and the stochastic decomposition feature was also explained. In [34], the author explored a single-server queueing system with geometric abandonments and server breakdowns for use in a multi-phase randomization scenario. According to this model, server breakdowns can only occur when the server is actively processing consumers.…”