Unreliable server with Non-Markovian Retrial Queueing System,Bernoulli Vacation and Fortuitous breakdown

Abstract

This paper explores the behaviour of a Bulk Arrival Retrial Queue Model (BARQ) with two phases of service under the Bernoulli Vacation schedule and Breakdown (BVSB). Each batch of customers arriving the system finds if the server is available, instantly utilizes the service. If the server is busy, under breakdown, or taking a vacation, then the customers enter into the orbit. After completing both service stages, the server will either take a vacation with probability p or wait until the next customer arrives with probability 1 - p or q. Our approach considers the nature of the customer as balking and also takes into account the breakdown of server, which may occur instantaneously during any stage of service. Significant performance measures have been derived and presented. A numerical study of the proposed model is carried out using MATLAB and results were reported.

Keywords

Retrial Queues two types of service Bernoulli Vacation steady-state Fortuitous Breakdown impatient customers

1 Introduction

In a retrial queue, the incoming calls which are directed to a busy server do not queue up or depart the situation instantly but rather move to a virtual location known as orbit and take their chance again after an arbitrary period of time [6, 15]. [16] considered an M/M/1 retrial queue with an unreliable server. [18] investigated a queue in the model that included retrials as well as individual failures. In addition, if the server is inactive when it suffers a passive-breakdown, the repair procedure will take longer since the cause of the problem cannot be pinpointed. On the other hand, if the server is in use when it suffers an active-breakdown, the process of repairing it will begin right away. They presented an interesting use of the paradigm in the PSN (Packet-Switching Network). The messages are slotted into TCP/IP (Transmission Control Protocol/Internet Protocol) packets before they are sent forward. These packets are then routed via a router forwarded from the source to the destination.

A repairable M/G/1 queueing model with setup times and N-policy was taken into consideration by [2]. In [9] discussed FRQB (Feedback Retrial Queue with Customer Balking) with server subject to breakdown. In their study of oscillating random walk models for the GI/G/1 vacation system in [10], the authors have discussed the topic of Bernoulli schedule discipline. [8] expressed the encouraging arrivals with feedback, balking, as well as maintaining reneged consumers. In [21], the authors introduced a model with the failure of servers and their repair process. The asymmetric randomized system failures and bulk arrivals of the BSVQ (Bernoulli schedule vacation queue) were explored in [7]. The vacation concept has been researched in theM/G/1 queueing system by numerous authors, including [11].

In [19], the author has categorized the flows in a network as elastic and inelastic which is followed by the scheduling of the elastic flows using DRR-SFF and the inelastic flows using BRR due to their huge requirements for capacity and delay constraints [3] explored the feasibility of a single server retrial queueing system, which includes optional re-service, client search, and delayed repair. In their most recent study, [1] investigated the feasibility of a two-phase heterogeneous service model. They started by optimizing an objective utilizing Canonical PSO, and then they constructed a bi-objective cost optimization model for the system while concurrently waiting for time. [22] pioneered the concept of a reserved server, subject to breakdown (with repairs and balking). By combining the concepts of balking, feedback, and subject to server breakdown and repair, we have enlarged the work of [4, 17] in this research.

In [20], the authors have developed a semi-analytic methodology and generated equations for the moment of the queue length in terms of server utilization in an M/G/1 structure with impatient subscribers. Discouraging behaviour in a queue indicates that customers are impatient when there is a long line, which is quite typical in many circumstances of congestion. In a batch arrival retrial queue including active breakdowns, [5] described how the server may indeed be immediately repaired when a breakdown occurs and how to take the interrupted customer’s reserved service schedule into consideration. The system has been reinforced with multi-phase optional services in addition to the essential services, and customers can choose from any of the optional services after receiving the essential services. Additionally, the system has been analyzed by [12] using the MEP (maximum entropy principle) to determine waiting times and steady-state solutions. Recently, [13] conducted analysis on a Markovian bulk service queue with feedback and SOS. The second service is the optional in RQS, and the BV (k types) were explored by [14].

The Retrial Queuing Model (RQM) with Bernoulli Vacation (BV) is the main focus of this study. In several past studies discussed above, a Single Server Retrial Queue Model (SSRQM) with two service phases and a Bernoulli vacation were discussed. Our proposed model describes an enhancement in SSRQM that includes two phases of service, under BV added with a balking state as well. The uniqueness of our approach comprises taking into consideration batch arrival, repair and instant breakdown under BV. We have developed vital performance metrics for our model, and we have also obtained the numerical investigation and graphical visualization of the model.

The rest of the paper follows the structure as Section 2 presents the mathematical model of our proposal. The governing equations of the model and the total number of consumers in the orbit/system are both calculated in Section 3. Section 4, offers a study of a variety of important performance metrics. The numerical findings and applications are discussed in Section 5 and Section 6. The concluding remarks with the future scope are been presented in Section 7.

2 Description of the proposed model

Let C_k to indicate the total number of clients belongs to the k^th arrival batches, according to a Poisson process with rate λ. Where C_k, k = 1, 2, 3,… are with a common distribution Pr [C_k = n] = χⁿ, n = 1, 2, 3… and C_(z) indicate the PGF of X.

The inter-retrial times have a random distribution U (υ_t) and a LST (Laplace-Stieltjes transform) U^* (θ) that corresponds to it. The function ϱ (υ_x) is the Hazard rates for repeated trials is given by, $ϱ (υ_{x}) d υ_{x} = \frac{U (υ_{x})}{1 - U (υ_{x})}$ .

The server’s vacation time has a random length V, distribution function V (υ_t) and LST V^* (θ). The functionϒ (υ_y) is the HRV (Hazard rates for vacation) is given by $ϒ (υ_{y}) d υ_{y} = \frac{V (υ_{y})}{1 - V (υ_{y})}$ .

Fig. 1

Pictorial representation of the model

Each individual consumer is given two distinct phases of heterogeneous service by the server. They are referred to as the initial phase service (Phase-I) and the later phase?service (Phase-II) respectively. The i^th phase service, i = 1, 2, with the d.f (distribution function)W_i (υ_t) and LST W^* (θ). The function Φ (υ_x) is that HRS (Hazard rates for services) is given by, $Φ_{i} (υ_{x}) d υ_{x} = \frac{W_{i} (υ_{x})}{1 - W_{i} (υ_{x})}$

Poisson processes with rates of α₁ for Phase-I and α₂ for Phase-II cause the breakdowns. The server’s repair time distributions (denoted by F₁ for Phase-I and F₂ for Phase-II) are assumed as being arbitrarily distributed with d.f. F₁ (υ_t) , F₂ (υ_t) and LST F^* (θ). The function Ψ (x) of the HRR (Hazard rates for repair) is given by, $Ψ_{i} (υ_{y}) d υ_{y} = \frac{F_{i} (υ_{y})}{1 - F_{i} (υ_{y})}$

In the steady state, we assume that U (0) =0, U (∞) =1, W_i (0) =0, W_i (∞) =1 are continuous at υ_x = 0 and V (0) =0, V (∞) =1, F_i (0) =0, F_i (∞) =1 are continuous at υ_y = 0 (for i = 1, 2).

Including random variables as well, and the server is $C (υ_{t}) = {\begin{matrix} 0 & \mapsto idle \\ 1 & \mapsto busy on Phase - I \\ 2 & \mapsto busy on Phase - II \\ 3 & \mapsto on vacation \\ 4 & \mapsto repair on Phase - I \\ 5 & \mapsto repair on phase - II \end{matrix} .$

3 Steady state distribution

We define the probabilities P₀ (υ_t) = P {C (υ_t) =0, N (υ_t) =0} and the probability densities for the process {N (υ_t) , υ_t ≥ 0} .

$\begin{matrix} Γ_{υ_{n}} (υ_{x}, υ_{t}) d υ_{x} = P {C (υ_{t}) = 0, N (υ_{t}) = υ_{n}, υ_{x} \leq U^{0} (υ_{t}) < υ_{x} + d υ_{x}}, for υ_{t} \geq 0, υ_{x} \geq 0, υ_{n} \geq 1 \end{matrix}$

$\begin{matrix} Δ_{1, υ_{n}} (υ_{x}, υ_{t}) d υ_{x} = P {C (υ_{t}) = 1, N (υ_{t}) = υ_{n}, υ_{x} \leq W_{1}^{0} (υ_{t}) < υ_{x} + d υ_{x}}, for υ_{t} \geq 0, υ_{x} \geq 0, υ_{n} \geq 0 \end{matrix}$ $\begin{matrix} Δ_{2, υ_{n}} (υ_{x}, υ_{t}) d υ_{x} = P {C (υ_{t}) = 2, N (υ_{t}) = υ_{n}, υ_{x} \leq W_{2}^{0} (υ_{x}) < υ_{x} + d υ_{x}}, for υ_{t} \geq 0, υ_{x} \geq 0, υ_{n} \geq 0 \end{matrix}$

$\begin{matrix} Λ_{1, υ_{n}} (υ_{x}, υ_{y}, υ_{t}) d υ_{y} = P {C (υ_{t}) = 3, N (υ_{t}) = υ_{n}, υ_{y} \leq F_{1}^{0} (υ_{t}) < υ_{y} + d υ_{y}}, for υ_{t} \geq 0, υ_{y} \geq 0, υ_{n} \geq 0 \end{matrix}$

$\begin{matrix} Λ_{2, υ_{n}} (υ_{x}, υ_{y}, υ_{t}) d υ_{y} = P {C (υ_{t}) = 4, N (υ_{t}) = υ_{n}, υ_{y} \leq F_{2}^{0} (υ_{t}) < υ_{y} + d y}, for t \geq 0, υ_{y} \geq 0, υ_{n} \geq 0 \end{matrix}$

$\begin{matrix} Ξ_{υ_{n}} (υ_{y}, υ_{t}) d υ_{y} = P {C (υ_{t}) = 5, N (υ_{t}) = υ_{n}, υ_{y} \leq V^{0} (υ_{t}) < υ_{y} + d υ_{y}}, for υ_{t} \geq 0, υ_{y} \geq 0, υ_{n} \geq 0 \end{matrix}$

In order to set, we assume that the sequel fulfills?the stability condition.

$\begin{matrix} L_{0} = lim_{n \to \infty} L_{0} (υ_{t}) for υ_{t} \geq 0 \end{matrix}$

$\begin{matrix} Γ_{υ_{n}} (υ_{x}) = lim_{υ_{n} \to \infty} Γ_{υ_{n}} (υ_{x}, υ_{t}) for υ_{t} \geq 0, υ_{x} \geq 0, υ_{n} \geq 1 \end{matrix}$

$\begin{matrix} Δ_{i, υ_{n}} (υ_{x}) = lim_{υ_{n} \to \infty} Δ_{i, υ_{n}} (υ_{x}, υ_{t}) for υ_{t} \geq 0, υ_{x} \geq 0, υ_{n} \geq 0 \end{matrix}$ $\begin{matrix} Λ_{i, υ_{n}} (υ_{x}, υ_{y}) = lim_{υ_{n} \to \infty} Λ_{i, υ_{n}} (υ_{x}, υ_{y}, υ_{t}) \\ for υ_{t} \geq 0, υ_{x} \geq 0, υ_{n} \geq 0 \end{matrix}$

$\begin{matrix} Ξ_{υ_{n}} (υ_{y}) = lim_{υ_{n} \to \infty} Ξ_{υ_{n}} (υ_{y}, υ_{t}) for υ_{t} \geq 0, υ_{n} \geq 0 \end{matrix}$

Following these, we have a set of equations that describes the dynamic behavior. $\begin{matrix} m λ L_{0} = \int_{0}^{\infty} Ξ_{0} (υ_{y}) ϒ (υ_{y}) d υ_{y} + q \int_{0}^{\infty} Δ_{2, 0} (υ_{x}) Φ_{2} (υ_{x}) d υ_{x} \end{matrix}$ (1)

$\begin{matrix} \frac{Ξ_{υ_{n}} (υ_{y})}{d υ_{y}} + [m λ + ϒ (υ_{y})] Λ_{i, υ_{n}} (υ_{x}, υ_{y}) = \\ m λ \sum_{k = 1}^{υ_{n}} χ_{k} Ξ_{i, υ_{n} - k} (υ_{x}, υ_{y}) \end{matrix}$ (2)

$\frac{Γ_{υ_{n}} (υ_{x})}{d υ_{x}} + [λ + ϱ (υ_{x})] Γ_{υ_{n}} (υ_{x}) = 0, υ_{n} \geq 1$ (3)

$\begin{matrix} \frac{Δ_{i, υ_{n}} (υ_{x})}{d υ_{x}} + [m λ + α_{i} + Φ_{i} (υ_{x})] Δ_{i, υ_{n}} (υ_{x}) = \\ m λ \sum_{k = 1}^{υ_{n}} χ_{k} Δ_{i, υ_{n} - k} (υ_{x}) + \int_{0}^{\infty} Ψ_{i} (υ_{y}) Λ_{i, υ_{n}} (υ_{x}, υ_{y}) d υ_{y}, \end{matrix}$ (4) $\begin{matrix} \frac{Λ_{i, υ_{n}} (υ_{x}, υ_{y})}{d υ_{y}} + [m λ + Ψ_{i} (υ_{y})] Λ_{i, υ_{n}} (υ_{x}, υ_{y}) = \\ m λ \sum_{k = 1}^{υ_{n}} χ_{k} Λ_{i, υ_{n} - k} (υ_{x}, υ_{y}) \end{matrix}$ (5)

The B.C (boundary conditions)are at υ_x = 0 and υ_y = 0 are $\begin{matrix} Γ_{υ_{n}} (0) = \int_{0}^{\infty} Ξ_{υ_{n}} (υ_{y}) ϒ (y) d υ_{y} + q \int_{0}^{\infty} Δ_{2, υ_{n}} (υ_{x}) Φ_{2} (υ_{x}) d υ_{x} \end{matrix}$ (6)

$\begin{matrix} Δ_{1, υ_{n}} (0) = \int_{0}^{\infty} Γ_{υ_{n} + 1} (υ_{x}) ϱ (υ_{x}) d υ_{x} \\ + λ \sum_{k = 1}^{υ_{n}} χ_{k} \int_{0}^{\infty} Γ_{υ_{n} - k + 1} (υ_{x}) d υ_{x} + m λ χ_{υ_{n} + 1} P_{0} \end{matrix}$ (7)

$Δ_{2, υ_{n}} (0) = \int_{0}^{\infty} Δ_{1, υ_{n}} (υ_{x}) Φ_{1} (υ_{x}) d υ_{x}$ (8) $Λ_{i, υ_{n}} (υ_{x}, 0) = α_{i} Δ_{i, υ_{n}}, υ_{n} \geq 1 for i = 1, 2$ (9) $Ξ_{υ_{n}} (0) = p \int_{0}^{\infty} Δ_{2, υ_{n}} (υ_{x}) Φ_{2} (υ_{x}) d υ_{x}$ (10)

The normalising condition is

$\begin{matrix} L_{0} + \sum_{υ_{n} = 1}^{\infty} \int_{0}^{\infty} Γ_{υ_{n}} (υ_{x}) d υ_{x} + \sum_{i = 1}^{2} {\sum_{υ_{n} = 0}^{\infty} \int_{0}^{\infty} Δ_{i, υ_{n}} (υ_{x}) d υ_{x} + \sum_{υ_{n} = 0}^{\infty} \int_{0}^{\infty} Λ_{i, υ_{n}} (υ_{x}, υ_{x}) d υ_{x} d υ_{y}} \\ + \sum_{υ_{n} = 0}^{\infty} \int_{0}^{\infty} Ξ_{υ_{n}} (υ_{y}) d υ_{y} = 1 \end{matrix}$ (11)

Now, multiplying equ(1 to 10) by υ_z and summing over υ_n, (υ_n = 0, 1, 2 …) $\frac{Γ (υ_{x}, υ_{z})}{d υ_{x}} + [λ + ϱ (υ_{x})] Γ (υ_{x}, υ_{z}) = 0, υ_{n} \geq 1$ (12)

$\begin{matrix} \frac{Δ_{1} (υ_{x}, υ_{z})}{d υ_{x}} + [m λ (1 - C (υ_{z})) + α_{1} + Φ_{1} (υ_{x})] Δ_{1} (υ_{x}, υ_{z}) = \int_{0}^{\infty} Ψ_{1} (υ_{y}) Λ_{1} (υ_{x}, υ_{y}, υ_{z}) d υ_{y}, \end{matrix}$ (13)

$\begin{matrix} \frac{Δ_{2} (υ_{x}, υ_{z})}{d υ_{x}} + [m λ (1 - C (υ_{x})) + α_{2} + Φ_{2} (υ_{x})] Δ_{2} (υ_{x}, υ_{z}) = \int_{0}^{\infty} Ψ_{2} (υ_{y}) Λ_{2} (υ_{x}, υ_{y}, υ_{z}) d υ_{y} \end{matrix}$ (14)

$\frac{Λ_{1} (υ_{x}, υ_{z})}{d υ_{x}} + [m λ (1 - C (υ_{z})) + Ψ_{1} (υ_{y})] Λ_{1} (υ_{x}, υ_{y}, υ_{z}) = 0$ (15)

$\frac{Λ_{2} (υ_{x}, υ_{z})}{d υ_{x}} + [m λ (1 - C (υ_{z})) + Ψ_{2} (υ_{y})] Λ_{2} (υ_{x}, υ_{y}, υ_{z}) = 0$ (16)

$\frac{Ξ (υ_{x}, υ_{z})}{d υ_{y}} + [m λ (1 - C (υ_{z})) + ϒ (υ_{y})] Ξ (υ_{x}, υ_{z}) = 0$ (17)

$\begin{matrix} Γ (0, υ_{z}) = \int_{0}^{\infty} Ξ (υ_{x}, υ_{z}) ϒ (υ_{y}) d υ_{y} + q \int_{0}^{\infty} Δ (υ_{x}, υ_{z}) Φ_{2} (υ_{x}) d υ_{x} - m λ L_{0} \end{matrix}$ (18)

$\begin{matrix} Δ_{1} (0, υ_{z}) = \frac{1}{υ_{z}} \int_{0}^{\infty} Γ (υ_{x}, υ_{z}) ϱ (υ_{x}) d υ_{x} + \frac{λ C (υ_{z})}{υ_{z}} {\int_{0}^{\infty} Γ (υ_{x}, υ_{z}) d υ_{x} + {mL}_{0}} \end{matrix}$ (19)

$Δ_{2} (0, υ_{z}) = \int_{0}^{\infty} Δ_{1} (υ_{x}, υ_{z}) Φ_{1} (υ_{x}) d υ_{x}$ (20) $Λ_{i} (υ_{x}, 0, υ_{z}) = α_{i} Δ_{i} (υ_{x}, υ_{z})$ (21) $Ξ (0, υ_{z}) = p \int_{0}^{\infty} Δ_{2} (υ_{x}, υ_{z}) Φ_{2} (υ_{x}) d υ_{x}$ (22) Solving equ (12) to (17), using partial differential equations, we obtain:

$Γ (υ_{x}, υ_{z}) = Γ (0, υ_{z}) (1 - U (υ_{x})) e^{- m λ υ_{x}}$ (23) $Δ_{i} (υ_{x}, υ_{z}) = Δ_{i} (0, υ_{z}) (1 - W_{i} (υ_{x})) e^{- ⊤_{i} (υ_{z}) υ_{x}}$ (24) $Λ_{i} (υ_{x}, υ_{y}, υ_{z}) = Λ_{i} (υ_{x}, 0, υ_{z}) (1 - F_{i} (υ_{y})) e^{- h (υ_{z}) υ_{y}}$ (25) $Ξ (υ_{x}, υ_{z}) = Ξ (0, υ_{z}) (1 - V (υ_{y})) e^{- h (υ_{z}) υ_{y}}$ (26)

where ⊤₁ (υ_z) = h (υ_z) + α₁ [1 - F₁ (h (υ_z))] , ⊤ ₂ (υ_z) = h (υ_z) + α₂ [1 - F₂ (h (υ_z))] and h (υ_z) = mλ (1 - C (υ_z)).

Applying equ (23) - (26) in (18) - (22) finally we get, Γ (0, υ_z) , Δ₁ (0, υ_z) , Δ₂ (0, υ_z) , Λ₁ (υ_x, υ_y, υ_z) , Λ₂ (υ_x, υ_y, υ_z) {and Ξ (0, υ_z). $Γ (0, υ_{z}) = \frac{m λ L_{0} [C (υ_{z}) (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) - υ_{z}]}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (27) $Δ_{1} (0, υ_{z}) = \frac{- L_{0} h (υ_{z}) U^{*} (λ)}{υ_{z} - [U^{*} (λ) + C (υ_{x}) (1 - U^{*} (λ))] (q + p V^{*} (h (υ_{x}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (28) $Δ_{2} (0, υ_{z}) = \frac{- L_{0} h (υ_{z}) U^{*} (λ) W_{1}^{*} (h (υ_{z}))}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (29) $Λ_{1} (υ_{x}, υ_{y}, υ_{z}) = \frac{- α_{1} L_{0} h (υ_{z}) (U^{*} (λ)) (1 - F_{1} (υ_{y})) e^{- ⊤_{1} (υ_{z}) υ_{x}} e^{- h (υ_{z}) υ_{y}}}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (30)

$Λ_{2} (υ_{x}, υ_{y}, υ_{z}) = \frac{- α_{2} L_{0} h (υ_{z}) (U^{*} (λ)) (1 - F_{2} (υ_{y})) e^{- ⊤_{2} (υ_{z}) υ_{x}} e^{- h (υ_{z}) υ_{y}}}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (31)

$Ξ (0, υ_{z}) = \frac{- {pL}_{0} h (υ_{z}) U^{*} (λ) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (32)

3.1 Theorem

Under the stability condition σ < 1, the stationary dist., of the no. of customers in the orbit when the server is empty, services busy on two phases, vacation, repair on two phases are given by

$Γ (υ_{z}) = \frac{m λ L_{0} (1 - U^{*} (λ)) [C (υ_{z}) (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) - υ_{z}]}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (33) $Δ_{1} (υ_{z}) = \frac{L_{0} h (υ_{z}) U^{*} (λ) (W_{1}^{*} (h (υ_{z})) - 1)}{z - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{x})) [⊤_{1} (υ_{z})]}$ (34) $Δ_{2} (υ_{z}) = \frac{L_{0} h (υ_{z}) U^{*} (λ) W_{1}^{*} (h (υ_{z})) (W_{2}^{*} (h (υ_{z})) - 1)}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) [⊤_{2} (υ_{z})]}$ (35) $Λ_{1} (υ_{z}) = \frac{α_{1} L_{0} (U^{*} (λ)) (1 - W_{1}^{*} (h (υ_{z}))) (F_{1} (h (υ_{z})) - 1)}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) [⊤_{1} (υ_{z})]}$ (36) $Λ_{2} (υ_{z}) = \frac{α_{2} L_{0} U^{*} (λ) (1 - W_{2}^{*} (h (υ_{z}))) (F_{2} (h (υ_{z})) - 1)}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) [⊤_{2} (υ_{z})]}$ (37)

$Ξ (υ_{z}) = \frac{{pL}_{0} U^{*} (λ) W_{1}^{*} (h (υ_{z})) W_{2} * (h (υ_{z})) (V^{*} (h (υ_{z})) - 1)}{υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))}$ (38)

where $L_{0} = {\frac{\begin{matrix} 1 - [I^{(1)} (1 - U^{*} (λ)) + m λ I^{(1)} [(p Ξ^{1} + (1 + α_{1} σ^{(1)}) δ^{(1)} + (1 + α_{2} σ^{(2)}) δ^{(2)})]] \end{matrix}}{\begin{matrix} 1 - m (1 - U^{*} (λ)) (1 - I^{(1)}) + (m - 1) (1 - U^{*} (λ)) \\ {[(p v^{1} + (1 + α_{1} σ^{(1)}) δ^{(1)} + (1 + α_{2} σ^{(2)}) δ^{(2)})] m (1 - U^{*} (λ))} \end{matrix}}}$ (39)

Proof:

Applying Γ (0, υ_z) , Δ₁ (0, υ_z) , Δ₂ (0, υ_z) , Λ₁ (υ_x, υ_y, υ_z) , Λ₂ (υ_x, υ_y, υ_z) {and Ξ (0, υ_z) values in (23) - (26) and integrating the partial probability generating functions from 0 to ∞ w.r (with respect)?to υ_x then we get

$Γ (υ_{z}) = \int_{0}^{\infty} Γ (υ_{x}, υ_{z}) d υ_{x} = Γ (0, υ_{z}) [1 - U (x)] e^{- λ υ_{x}}, Δ_{i} (υ_{z}) = \int_{0}^{\infty} \int_{0}^{\infty} Δ_{i} (υ_{x}, υ_{z}) d υ_{x}, Λ_{i} (υ_{x}, υ_{z}) = \int_{0}^{\infty} Λ_{i} (υ_{x}, υ_{y}, υ_{z}) dy, Λ_{i} (υ_{z}) = \int_{0}^{\infty} Λ_{i} (υ_{x}, υ_{z}) d υ_{x}, Ξ (υ_{z}) = \int_{0}^{\infty} Ξ (υ_{x}, υ_{x}) d υ_{x}$ .

Consequently, using the normalization condition L₀, By choosing υ_z = 1, we can calculate the prob., that the server is empty when there are no consumers in the orbit in equ. (33) to (38) and by applying L-Hospital rule we get L₀ + Δ₁ (1) + Δ₂ (1) + Λ₁ (1) + Λ₂ (1) + Ξ (1) =1.

3.2 Corollary

The PGF of the number of customer in the orbital is and The PGF of the number of customer in the system is: H (υ_z) = L₀ + Γ (υ_z) + Δ₁ (υ_z) + Δ₂ (υ_z) + Λ₁ (υ_z) + Λ₂ (υ_z) + Ξ (υ_z)

K (υ_z) = L₀ + Γ (z) + Ξ (υ_z) + υ_z [Δ₁ (υ_z) + Δ₂ (υ_z) + Λ₁ (υ_z) + Λ₂ (υ_z)]

$H (υ_{z}) = L_{0} {\frac{\begin{matrix} (υ_{z} (1 - m) + U^{*} (λ) (m υ_{z} - 1) + (m - 1) C (υ_{z}) (1 - U^{*} (λ)) \\ (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z}))) \end{matrix}}{\begin{matrix} υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) \end{matrix}}}$ (40) $K (υ_{z}) = L_{0} {\frac{\begin{matrix} (υ_{z} (1 - U^{*} (λ)) (1 - m) + (υ_{z} - 1) U^{*} (λ) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) + \\ (m - 1) C (υ_{z}) (1 - U^{*} (λ)) (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2} * (h (υ_{z}))) \end{matrix}}{\begin{matrix} υ_{z} - [U^{*} (λ) + C (υ_{z}) (1 - U^{*} (λ))] (q + {pV}^{*} (h (υ_{z}))) W_{1}^{*} (h (υ_{z})) W_{2}^{*} (h (υ_{z})) \end{matrix}}}$ (41)

4 Performance Measures

4.1 Theorem

We obtain the following probability if the system satisfies the stability condition: where ${Dr}^{'} = {\begin{matrix} 1 - {I^{(1)} (1 - U^{*} (λ)) + \\ λ {mI}^{(1)} [{pv}^{1} + (1 + α_{1} σ^{(1)}) δ^{(1)} \\ + (1 + α_{2} σ^{(2)}) δ^{(2)}]} \end{matrix}}$

(a) The probability that its server will be inactive during retrial

$Γ (1) = {mL}_{0} (1 - U^{*} (λ)) {\frac{\begin{matrix} {I^{(1)} [1 + m λ (p Ξ^{1} + \\ (1 + α_{1} σ^{(1)}) δ^{(1)} + \\ (1 + α_{2} σ^{(2)}) δ^{(2)})] - 1} \end{matrix}}{{Dr}^{'}}}$ (42)

(b) The probability that the server will be busy on Phases I and II:

$Δ (1) = L_{0} {\frac{m λ U^{*} (λ) I^{(1)} δ^{(1)}}{{Dr}^{'}}}$ (43) $Δ (2) = L_{0} {\frac{U^{*} (λ) I^{(1)} m λ δ^{(2)}}{{Dr}^{'}}}$ (44)

$Λ_{(1)} = L_{0} {\frac{α_{1} m λ U^{*} (λ) I^{(1)} δ^{(1)} σ^{(1)}}{{Dr}^{'}}}$ (45) $Λ_{(2)} = L_{0} {\frac{α_{2} U^{*} (λ) I^{(1)} m λ δ^{(2)} σ^{(2)}}{{Dr}^{'}}}$ (46)

(d) The Probability that the server vacation state: $Ξ (1) = L_{0} {\frac{m λ p U^{*} (λ) v^{(1)} I^{(1)}}{{Dr}^{'}}}$ (47)Proof: Values in equations (33) to (38), that we have already proved, are then substituted by υ_z = 1

ie., $Γ (1) = lim_{υ_{z} \to 1} Γ (υ_{z}), Δ (1) = lim_{υ_{z} \to 1} Δ_{1} (υ_{z}), Δ (2) = lim_{υ_{z} \to 1} Δ_{2} (υ_{z}), Λ (1) = lim_{υ_{z} \to 1} Λ_{1} (υ_{z}), Λ (2) = lim_{υ_{z} \to 1} Λ_{2} (υ_{z}), Ξ (1) = lim_{υ_{z} \to 1} Ξ (υ_{z})$ . By doing a simple computation, the formula is followed.

4.2 Corollary

(a) By assessing υ_z = 1 and differentiating (41) w.r. to υ_z, the expected number of customers in the system is determined.

$L_{s} = \frac{L_{0} {{Nr}_{k}^{″} {Dr}^{'} - {Dr}^{″} {Nr}_{k}^{'}}}{2 V^{*} (λ) ({Dr}^{'^{2}})}$ (48)

(b) By assessing υ_z = 1 and differentiating (40) with respect to υ_z, the expected number of customers in the orbit may is determined. $L_{q} = \frac{L_{0} {{Nr}_{h}^{″} {Dr}^{'} - {Dr}^{″} {Nr}_{h}^{'}}}{2 V^{*} (λ) ({Dr}^{'^{2}})}$ (49)

(c) The steady-state average time a customer spends in the system (W_s) and orbit (W_q) are calculated using Little’s formula $W_{s} = \frac{L_{s}}{λ}$ and $W_{q} = \frac{L_{q}}{λ}$ .

5 Numerical Examples

This section provides a numerical representation of our model’s real-world application, which is the IEM (Internet email system)?utilized in computer networking. Emails get sent from a starting point to a destination using this system. We show how numerous variables affect the system performance statistic, which is spread exponentially overall?retry times, service times, BV times and repair times. Tables 2?applies Φ₁ = 4, Φ₂ = 5, p = 3, m = 0.4, α₁ = 6, α₂ = 7 for the arbitrary chosen values of λ= 0.8. If the vacation rate is growing, Lo and BV are likewise rising, while Lq is decreasing, as seen in Table 1. Table 2 demonstrates that when the retry rate rises, L_o, Δ (1) and Δ (2) are also growing.

Table 1
The effect of vacation rate (ϒ) on L₀ and Ξ

ϒ L ₀ Ξ L _q

2.1 0.1867 0.3904 0.5481

2.3 0.2363 0.5413 0.3772

2.5 0.2779 0.6923 0.2884

2.7 0.3134 0.8433 0.2322

2.9 0.3440 0.9945 0.1956

ϒ	L ₀	Ξ	L _q
2.1	0.1867	0.3904	0.5481
2.3	0.2363	0.5413	0.3772
2.5	0.2779	0.6923	0.2884
2.7	0.3134	0.8433	0.2322
2.9	0.3440	0.9945	0.1956

Table 2

The effect of retry rate (ϱ) on L₀, Δ (1) and Δ (2)

ϱ	L ₀	Δ (1)	Δ (2)
15	0.8030	0.0251	0.0602
17	0.8100	0.0255	0.0612
19	0.8156	0.0258	0.0620
21	0.8201	0.0261	0.0626
23	0.8239	0.0263	0.0632

In Fig.2 and Fig.3, the rise in the value of ϱ (retry rate) and ϒ (vacation rate) leads to an increase in the idle prob.,(L₀). The figures labeled 4 and 5 each illustrates a graph in three dimensions. In Fig.4, ϱ and ϒ escalates, L₀ also escalates, and in Fig.5, ϒ and Φ₁ escalates, L₀ also escalates.

Fig. 2

L₀ versus ϱ and ϒ.

Fig. 3

L₀ versus Retrial rate and Vacation rate.

Fig. 4

L₀ versus ϱ and ϒ.

Fig. 5

L₀ versus ϒ and Φ₁.

6 Application of the model

Virus scanning is a necessary and useful tool for email server maintenance since virus attacks have a serious impact on the email service system. These virus attacks are a analogy to the system breakdown state. The email server does not handle any email when the system is in vacation state (when exposed to the virus) or breakdown state and hence the emails are moved to orbit. If any virus is found then the next vacation will start for the scanning of virus. If virus scans are only carried out when the server is in vacation, on the other hand, the system may be at a high risk of attack because of a prolonged busy time. It is possible to reduce this risk by carrying out random scans after a service is complete. In essence, after the service is complete, the server either handles the next email with probability 1 - p, if the system is not empty or does a virus check with probability p, where p is known as the vacation probability.

7 Conclusion

In this paper, we investigated the breakdown and delay periods of an M^X/G/1 queue with general retrial time, two phases of service and Bernoulli vacation schedule for an unreliable server and impatient customers. The concept has several uses in fields like Packet-Switched Networks, Local Area Networks, Manufacturing Lines, Client-Server Communication, Mobile Systems, Call Centres, and Internet Email System has been presented for the significant performance metrics. Our future work may be focused on retrial queues with interruptions, catastrophes, orbital search, priority,various vacation principles, and feedback to optimize the current queueing model.

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Unreliable server with Non-Markovian Retrial Queueing System,Bernoulli Vacation and Fortuitous breakdown

Abstract

Keywords

1 Introduction

2 Description of the proposed model

4.1 Theorem

Table 1 The effect of vacation rate (ϒ) on L0 and Ξ ϒ L 0 Ξ L q 2.1 0.1867 0.3904 0.5481 2.3 0.2363 0.5413 0.3772 2.5 0.2779 0.6923 0.2884 2.7 0.3134 0.8433 0.2322 2.9 0.3440 0.9945 0.1956

7 Conclusion

References

Table 1
The effect of vacation rate (ϒ) on L₀ and Ξ

ϒ L ₀ Ξ L _q

2.1 0.1867 0.3904 0.5481

2.3 0.2363 0.5413 0.3772

2.5 0.2779 0.6923 0.2884

2.7 0.3134 0.8433 0.2322

2.9 0.3440 0.9945 0.1956