Multiserver queue with addressed retrials

Abstract

This paper presents a multiserver retrial queueing system with servers kept apart, thereby rendering it impossible for one to know the status (idle/busy) of the others. Customers proceeding to one channel will have to go to orbit if the server in it is busy and retry after some time to some channel, not necessarily the one already tried. Each orbital customer, independently of others, chooses the server randomly according to some specified probability distribution. Further this distribution is identical for all customers. We assume that the same 'orbit' is used by all retrial customers, between repeated attempts, to access the servers. We derive the system state probability distribution under Poisson arrival process of external customers, exponentially distributed service times and linear retrial rates to access the servers. Several system state characteristics are obtained and numerical illustrations provided.

Analysis of retrial queue with heterogeneous servers and Markovian arrival process

“…Многолинейные системы с повторными вызовами, в которых серверы являются однородными, а вновь прибывающий запрос выбирает произвольный незанятый прибор с равной вероятностью и обращается к какому-то конкретному прибору, рассматривались, например, в работах [16,17].…”

Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory

“…Continuous time AQTMC were applied to investigate the queueing model BMAP/PH/N/0 in the case of complete admission discipline, see [24], the M/M/c retrial queue with controllable number of active servers, see [2], the M/M/c retrial queue with addressed retrials where the primary or repeated customer is addressed, in each attempt, to a concrete server, but not to a whole pool of servers, see [31].…”

Stability of the multiserver queue with addressed retrials

“…The service time distribution depends on the server selected and for rth server the distribution is exponential with the same parameter μ r both for primary and returning customers. This assumption is more general as compared with Mushko et al (2006) where it is assumed that μ r ≡ μ. Note that the model under consideration for the first time was studied in Mushko (2006).…”

“…When all probabilities θ r are positive, i.e. returning customers have access to any server, and μ r ≡ μ it is shown in Mushko et al (2006) that the process ζ(t) is ergodic if λ < cμ (as a matter of fact this inequality is also necessary for the ergodicity). The situation when some servers are reserved to process primary customers, i.e.…”

“…the corresponding probabilities θ k = 0, is mentioned as an interesting case which requires special consideration. The authors of Mushko et al (2006) guessed that in that case the stability condition depends on the distribution q k , but did not even write out the condition in an explicit form.…”