Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance
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Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance Srinivas R. Chakravarthy1 · Shruti2 · Alexander Rumyantsev3,4 Received: 16 January 2020 / Revised: 23 September 2020 / Accepted: 28 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call group clearance. The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type. Keywords Queueing · Stochastic processes · Batch arrivals · Group clearance · Matrix-analytic method Mathematics Subject Classification (2010) 60J27 · 60K25
1 Introduction Batching (bulking) and group clearance are natural ways to improve throughput and utilization of a system used in various fields from public transportation (Grippa et al. 2019) and Srinivas R. Chakravarthy
[email protected] 1
Departments of Industrial and Manufacturing Engineering, Mathematics, Kettering University, Flint, MI 48504, USA
2
Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Pilani, Rajasthan 333031, India
3
Institute of Applied Mathematical Research, Karelian Research Centre of RAS, 11 Pushkinskaya Str., Petrozavodsk, Russia
4
Petrozavodsk State University, 33 Lenina Pr., Petrozavodsk, Russia
Methodology and Computing in Applied Probability
cargo to blood screening, production systems, amusement parks etc. (a number of interesting applications with appropriate references can be found in Claeys et al. (2013)). In information technology batching is widely implemented in telecommunication and computing systems. For instance, packages in most common implementations of the TCP protocol are grouped into the so-called windows, which are sent to a receiver simultaneously. Another example is the packet encapsulation during transmission, when aggregates rather than standalone packages, equipped with a single header, are sent in a single transmission. The effect of increased throughput along with improved channel utilization by frame aggregation plays a crucial role in recent 802.11 standards and is further amplified in the case of IEEE 802.11n WLANs (Charfi et al. 2017). By the same reason of improving the throughput and reducing the late
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