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Queue occupancy and server size distribution of a queue length dependent vacation queue with an optional service

Ashish Verma, Sourav Pradhan

TL;DR

This article analyzes an infinite-buffer discrete-time batch-arrival queue with single and multiple vacation policy where customers are served in batches, in two phases, namely first essential service (FES) and second optional service (SOS).

Abstract

The discrete time queueing system is highly applicable to modern telecommunication systems, where it provides adaptive packet handling, congestion controlled security/inspection, energy efficient operation, and supports bursty traffic common in 5G, Internet of Things (IoT), and edge computing environments. In this article, we analyze an infinite-buffer discrete-time batch-arrival queue with single and multiple vacation policy where customers are served in batches, in two phases, namely first essential service (FES) and second optional service (SOS). In such systems, the FES corresponds to basic data processing or packet routing, while SOS represents secondary tasks such as encryption, error checking, data compression, or deep packet inspection that may not be necessary for every packet. Here, we derive the bivariate probability generating functions for the joint distribution of the number of packets waiting for transmission and the number are being processed immediately after the completion of both the FES and SOS. Furthermore, the complete joint distribution at arbitrary time slots, including vacation completion states, is established. Numerical illustrations demonstrate the applicability of the proposed framework, including an example with discrete phase type service time distribution. Finally, the sensitivity analysis of the key parameters on marginal system's probabilities and different performance measures have been investigated through several graphical representations.

Queue occupancy and server size distribution of a queue length dependent vacation queue with an optional service

TL;DR

This article analyzes an infinite-buffer discrete-time batch-arrival queue with single and multiple vacation policy where customers are served in batches, in two phases, namely first essential service (FES) and second optional service (SOS).

Abstract

The discrete time queueing system is highly applicable to modern telecommunication systems, where it provides adaptive packet handling, congestion controlled security/inspection, energy efficient operation, and supports bursty traffic common in 5G, Internet of Things (IoT), and edge computing environments. In this article, we analyze an infinite-buffer discrete-time batch-arrival queue with single and multiple vacation policy where customers are served in batches, in two phases, namely first essential service (FES) and second optional service (SOS). In such systems, the FES corresponds to basic data processing or packet routing, while SOS represents secondary tasks such as encryption, error checking, data compression, or deep packet inspection that may not be necessary for every packet. Here, we derive the bivariate probability generating functions for the joint distribution of the number of packets waiting for transmission and the number are being processed immediately after the completion of both the FES and SOS. Furthermore, the complete joint distribution at arbitrary time slots, including vacation completion states, is established. Numerical illustrations demonstrate the applicability of the proposed framework, including an example with discrete phase type service time distribution. Finally, the sensitivity analysis of the key parameters on marginal system's probabilities and different performance measures have been investigated through several graphical representations.
Paper Structure (23 sections, 1 theorem, 41 equations, 16 figures, 6 tables)

This paper contains 23 sections, 1 theorem, 41 equations, 16 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Literature review
  3. Motivation and major contribution
  4. Organization of the paper
  5. Description of proposed mathematical model
  6. Applications
  7. Application of the model in Wireless Networks / IoT / Cloud computing/ Manufacturing
  8. Other areas of possible application of the model
  9. State of the system and difference equations
  10. Stationary joint distribution at service/vacation completion epoch
  11. Bivariate pgfs
  12. Extraction of joint probabilities
  13. Particular cases and matching with existing results
  14. Stationary joint distribution at arbitrary slot
  15. Performance indices
  16. ...and 8 more sections

Key Result

Theorem 7.1

The joint probabilities at service completion and at an arbitrary slot i.e., $(\alpha_{n,r},\alpha_{n,r}^{+},\alpha_n^{+}),$$(\beta_{n,r}, \beta_{n,r}^{+}, \beta_n^{+})$ and $(\gamma_{n}^{[k]},\gamma_{n}^{[k]+})$are connected through the following relation: where

Figures (16)

  • Figure 2: For single vacation
  • Figure 3: For multiple vacations
  • Figure 4: For single vacation
  • Figure 5: For multiple vacations
  • Figure 6: For single vacation
  • ...and 11 more figures

Theorems & Definitions (3)

  • proof
  • Theorem 7.1
  • proof