Distribution of the number of losses in busy-periods of oscillating systems
Introduction
The traditional queues are single server systems at which customers arrive in batches of random size (X), and that allow a maximum capacity of n users simultaneously in the system. In these systems, the interarrival times are assumed to be independent and exponentially distributed (M) and the service times follow a general distribution (G) that remains the same regardless of the number of customers in the system. However, when managing real life queueing systems, in particular when the server is a human being, such homogeneous service assumption is usually inappropriate as the queue length perception frequently affects the server’s productivity. In fact, as servers realise that a large load in the system can lead to a strong degradation in the quality of service delivered, a server that sees many customers waiting for service will tend to increase its service rate. In turn, whenever a server has hardly any work to do he/ she tends to relax and slow down the service rate (cf., e.g., Delasay et al., 2019, Do et al., 2015).
Mobile and high speed telecommunication networks is another context in which it is common to consider queueing systems that adapt their service characteristics to the network congestion conditions (cf., e.g., Altman et al., 2002, Choi et al., 2001, Choi and Choi, 1996, Choi et al., 1999, Gupta et al., 2014, Sriram and Lucantoni, 1989, Sriram et al., 1991 and the references therein).
As previously stated, the service rate of the classical queueing systems is constant. Therefore, these systems are not tailored to jointly achieve short mean customer waiting times and high server utilization, as high utilization of the server corresponds to high traffic intensity and congestion of the system, leading to loss of service quality. As a consequence, there is a strong practical interest in investigating oscillating queueing systems in which the customer service times are not independent of each other, but instead depend on the evolution of the number of customers in the system.
In the paper, we consider single server oscillating systems under a threshold policy with hysteresis. The motivation for using a threshold-based approach is the improvement of the system cost/performance ratio. Specifically, under light loads, it is not desirable that the system operates with an unnecessarily fast server, due to its incurred usage costs. On the other hand, it is also not desirable for a system to exhibit very long delays, which can result from the use of a slow (less costly) server under heavy loads.
By dynamically changing the type of service offered, depending on the system load, one can maintain an acceptable expected service time (i.e. not too long) while achieving a reasonable (i.e. not too high) cost for operating the system. However, it is not desirable to constantly interchange between faster and slower service to react to instantaneous changes/fluctuations in the workload around a single threshold value. It is desirable to offer a faster (more costly) service only when the system is moving towards a heavy loaded operation scenario, and to change to a slower service only when the system is moving towards a light loaded operation scenarion. Such oscillations are avoided considering in the system a (upward) threshold b and hysteresis (i.e., a downward threshold, a). Similarly, in the spirit of improving the cost/performance ratio of the system, reacting to changes in workload through the use of thresholds, oscillating systems differ from other threshold queues with hysteresis proposed in the literature (see, e.g., Golubchik and Lui, 2002, Sriram and Lucantoni, 1989, Yadin and Naor, 1967, and references therein) in the fact that they maintain a single server permanently.
In this paper we propose a recursive procedure to compute the probability function of the number of customer losses during busy-periods of preemptive and non-preemptive oscillating queues, extending the results for classical systems derived in Ferreira et al. (2013). The analysis of customer losses during busy-periods is relevant to compute important performance measures such as customer loss rates in busy-periods and long-run blocking probabilities (Ferreira et al., 2016). In general, the relevance of characteristics associated to busy-periods for finite capacity Markovian arrival queues is well reflected by the abundance of studies of such characteristics in the literature (see, e.g., Abramov, 1997, Abramov, 2011, Al Hanbali and Boxma, 2010, Ferreira et al., 2013, Pacheco and Ribeiro, 2006, Pacheco and Ribeiro, 2008, Peköz, 1999, Peköz et al., 2003, Righter, 1999, Wolff, 2002 and the references therein).
The remainder of the paper is organized as follows. In Section 2 we describe the studied preemptive and non-preemptive systems, and in Section 3 we propose a recursive scheme to compute the probability function for the number of customer losses in busy-periods of such systems. An algorithmic implementation of the recursive scheme is proposed in Section 4. Finally, we present numerical results that illustrate the effectiveness of the proposed method in Section 5 and draw some conclusions in Section 6. The Appendix gives details on how to compute the mixed-Poisson probabilities, needed for the algorithmic implementation, for specific service time distributions.
Section snippets
Model description
Oscillating systems are single server queueing systems with finite capacity n, including the customer in service – if any, and thresholds a and b. The customers arrive in batches according to a compound Poisson process with rate . The sequences of batch sizes and batch interarrival times are independent. The batch sizes, , are independent and identically distributed to a random variable X having probability function , with finite mean , where . The
Number of customer losses in a busy-period
In this section, we address the probability distribution of the number of customers lost in busy-periods of oscillating systems, extending the results for systems presented in Ferreira et al. (2013). By busy-period we mean the period of time that starts when a batch of customers arrives to an empty system and ends at the first subsequent time the system becomes empty. Therefore, we consider multi busy-periods,i.e., busy-periods initiated with multiple customers in the
Algorithmic implementation
In this section, we propose a generic algorithm to compute the distribution of the number of customers lost in -busy-periods of oscillating systems.
Before proceeding, we start by noting that conditioning on the number of batches that arrives during a random amount of time with distribution A, the probability that exactly l customers arrive during that random amount of time, , can be computed from cf. Ferreira et al. (2017)where, for
Numerical illustration
The algorithm presented in Fig. 1 has been implemented in MATLAB to compute the probability function of the number of customer lost in busy-periods of preemptive oscillating systems, in the following denoted by systems, with . In particular, we illustrate the sensitivity of the distribution of the number of customer losses in busy-periods with respect to the traffic intensity, the service distributions, and group size distribution. To this
Conclusion
In the paper we investigate the probability function of the number of customer losses in busy-periods of preemptive and non-preemptive oscillating systems. The main result consists of a recursive procedure to compute such probability function that enables a quick calculation of the customer loss probabilities, extending the results for classical systems derived in Ferreira et al. (2013). These systems include as special cases, among others, the oscillating
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement
Fátima Ferreira: Conceptualization, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. António Pacheco: Conceptualization, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. Helena Ribeiro: Conceptualization, Investigation, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing.
Acknowledgements
This research was partially supported by Fundação para a Ciência e a Tecnologia (FCT) through project UID/Multi/04621/2019.
References (31)
- et al.
Busy period analysis of the state dependent M/M/1/K queue
Oper. Res. Lett.
(2010) - et al.
State-dependent M/G/1 type queueing analysis for congestion control in data networks
Comput. Netw.
(2002) The oscillating queue with finite buffer
Perf. Eval.
(2004)- et al.
Load effect on service times
Eur. J. Oper. Res.
(2019) - et al.
Analysis of a discrete-time queue with load dependent service under discrete-time markovian arrival process
J. Korean Stat. Soc.
(2014) - et al.
Critical starting points for stable evaluation of mixed-Poisson probabilities
Math. Econ.
(1993) On a property of a refusals stream
J. Appl. Prob.
(1997)Statistical analysis of single-server loss queueing systems
Method. Comput. Appl. Prob.
(2011)Busy period analysis of the level dependent PH/PH/1/K queue
Queueing Syst.
(2011)- et al.
The queue with queue length dependent service times
J. Appl. Math. Stoch. Anal.
(2001)