Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 305 https://internationalpubls.com A Single Server Markovian General Service Retrial- Encouraged arrival queuing system with Persistent Retrial Technique under the Transitory Behaviour Ismailkhan E1*, S. Moorthy2, P. Thangaraja3*, Syed Siddiqua Begum4 and K. Rajesh5 1,3 Department of Mathematics, Panimalar Engineering College, Chennai-600 123, Tamil Nadu, India. 2,5 Department of Mathematics, Government Arts and Science College for Women, Bargur-635104, Tamil Nadu, India. 4 Department of Mathematics and Actuarial Science, B.S. Abdur Rahman Crescent Institute of Science and Technology, Vandalur, Chennai, Tamil Nadu, India. Corresponding Author e-mail: ismailkhanmubarak@gmail.com, thangarajap1991@gmail.com Article History: Received: 01-08-2024 Revised: 09-09-2024 Accepted: 19-09-2024 Abstract: In this paper, the objective of this research is to analyse the behaviour and performance of a single server retrial-encouraged arrival queuing system. Specifically, it aims to derive the steady-state distributions for the system using the transitory technique. To calculate the performance metrics such as the average number of consumers in the orbit, and probabilities of the server being busy and idle. Hereafter conduct the numerical illustration for various service time distributions like (Exponential and Erlang) to understand the impact of different system performances on parameters. This research helps in understanding how the system performs under different conditions and can be useful for optimizing queue management in different applications. Keywords: Encouraged arrival, Retrial Technique, Transitory behaviour, Exponential distribution, Erlangian distribution 1. Introduction Retrial queues are queuing systems that allow incoming customers to retry for service after a certain amount of time if they discover the server and any waiting spots to be occupied. An essential component of the concept of queueing is the model of retrial queues. The need to account for the retry effect in everyday life and different networking systems gives birth to these models. For this reason, the study of such queue models receives a lot of interest. Several communication networks can be accurately described by trial queueing models. Their inquiry is, therefore, crucial. In tele-traffic theory and telephone networks where customers redial after receiving an engaged signal, retry queues have been seen as an intriguing subject. A comparative analysis of standard queuing systems and the retrial queuing technique is investigated in [1]. A comprehensive and modern treatment of retrial queuing systems through computational techniques was studied in [2]. The stochastic processes arising from these models in stationary and non-stationary regimes are investigated in [3]. Conduct the single-server queue under demands examined in [4]. A retrial queuing system with results was studied in [5]. The primary models and results in the field of retrial queues are investigated in [6]. mailto:ismailkhanmubarak@gmail.com mailto:thangarajap1991@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 306 https://internationalpubls.com A comprehensive overview of the main results and methods in the theory of retrial queues studied in [7]. [8] Examines the steady-state behaviour of an M/G/1 queueing system where the server may provide two phases of heterogeneous service to incoming units. [9] Investigate the steady-state behaviour of an M/G/1 queueing system that incorporates two phases of service and operates under a D-policy. The steady-state behaviour of an M/G/1 queueing system that incorporates repeated attempts and two-phase service was studied in [10]. [11] Studied the behaviour of an M/G/1 queuing system that adds a second optional service. [12] analyzes an M/G/1 queueing system that incorporates a second optional service and server vacations scheduled according to a Bernoulli process. Investigating an M/M/1/N system with encouraged arrivals was done in [13,14]. Studying an M/M/1 retrial queuing system was done in [15] 2. Model Assumptions: The following presumptions form the basis of this model's analysis. • Patron entry follows an Encouraged arrival with rate Λ × (1 + δ). δ represents the discount values are 10% i.e The interval between the Encouraged arrival is an identical distribution under an average rate of 1 Λ×(1+δ) . The service time adheres to a General distribution under a conditional probability density of service via the interval (𝑦, 𝑦 + Δ ∗ 𝑦) to obtain the expected service time 𝑦, we have 𝜇(𝑦) = 𝑎(𝑦) 1 − 𝐴(𝑦) (1) where 𝑎(𝑦) = 𝜇(𝑦)𝑒−∫   𝑦 0  𝜇(𝑇)𝑑𝑇 (2) • Patron follows FCFS discipline • The extend of the patron and group is infinite, beginning with arrival via the inter of time is (𝑇, 𝑇 + Δ ∗ 𝑇) is Λ × (1 + δ)Δ𝑇 + 𝑧(Δ ∗ 𝑇). Hereafter, retry attempts via the time of interval is (𝑇, 𝑇 + Δ ∗ 𝑇) is obtained that 𝜒Δ ∗ T + z(Δ ∗ t). Multiple departures within the given time interval (𝑇, 𝑇 + Δ ∗ 𝑇) are 0. • The retrial rate is (1 − 𝜓zm)𝜒, where 𝜓zm denotes Kronecker's delta. • The transitory behaviour of this model is developed by the Supplementary variable technique. • Steady-state of the probability distributions derived for different service-time distributions (Exponential and Erlang). The probability of the system-server being idle engages for different parameters. 3. To Derive the Set of Differential-Difference Equations of the Model We write the basic equations and derive the governing equations are described as follows 𝑃𝑚1 ′ (𝑦, 𝑇) + ∂ ∂𝑇 𝑃𝑚1(𝑦, 𝑇) + (Λ × (1 + δ) + 𝜇(𝑦))𝑃𝑚1(𝑦, 𝑇) = Λ × (1 + δ)𝑃𝑚−11(𝑦, 𝑇) (3) 𝑃01 ′ (𝑦, 𝑇) + ∂ ∂𝑇 𝑃01(𝑦, 𝑇) + (Λ × (1 + δ) + 𝜇(𝑦))𝑃01(𝑦, 𝑇) = 0 (4) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 307 https://internationalpubls.com 𝑃𝑚0 ′ (𝑇) + (Λ × (1 + δ) + 𝜎)𝑃𝑚0(𝑇) = ∫ 0 ∞  𝑃𝑚1(𝑦, 𝑇)𝜇(𝑦)𝑑𝑦 forn = 1,2,3, … (5) 𝑃00 ′ (𝑇) + Λ × (1 + δ)𝑃00(𝑇) = ∫0 ∞  𝑃01(𝑦, 𝑇)𝜇(𝑦)𝑑𝑦 = ∫0 ∞  𝑃01(𝑦, 𝑇)𝜇(𝑦)𝑑𝑦 (6) The subsequent boundary conditions apply while solving the aforementioned equations (3) through (6) 𝑃𝑚1(0, 𝑇) = Λ × (1 + δ)𝑃𝑚0(𝑇) + 𝜎𝑃𝑚+10(𝑇) (7) 𝑃01(0, 𝑇) = Λ × (1 + δ)𝑃00(𝑇) + 𝜎𝑃10(𝑇) (8) Primary condition is 𝑃00(0) = 1 (9) For the server that is engaged or unoccupied in the transitory state, we create the subsequent probability-generating functions 𝑃0(𝑇, 𝑍) = ∑ 𝑚=0 ∞  𝑃𝑚0(𝑇)𝑍 𝑚and𝑃1(𝑇, 𝑍) = ∑𝑚=0 ∞  𝑃𝑚1(𝑇)𝑍 𝑚 (10) The LT of 𝑔(𝑇) is described by 𝑔∗(𝑐) = ∫ 0 ∞  𝑒−𝑐𝑇𝑔(𝑇)𝑑𝑇 (11) Apply LT equations (3) to (10), we obtain 𝑃′𝑚1 ∗ (𝑦, 𝑐) + (𝑐 + Λ × (1 + δ) + 𝜇(𝑦))𝑃𝑚1 ′∗ (𝑦, 𝑐) = Λ × (1 + δ)𝑃′𝑚−11 ∗ (𝑦, 𝑐) for m = 1,2,3, … (12) 𝑃′01 ∗ (𝑦, 𝑐) + (𝑐 + Λ × (1 + δ) + 𝜇(𝑦))𝑃′01 ∗ (𝑦, 𝑐) = 0 (13) (𝑐 + Λ × (1 + δ) + 𝜎)𝑃𝑚0 ∗(𝑇) = ∫ 0 ∞  𝑃𝑚1 ∗(𝑦, 𝑐)𝜇(𝑦)𝑑𝑦 for m = 1,2,3, … (14) (Λ × (1 + δ) + 𝑐)𝑃00 ∗(𝑐) = 1 + ∫ 0 ∞  𝑃01 ∗(𝑦, 𝑐)𝜇(𝑦)𝑑𝑦 (15) The subsequent equations (12) to (15) are derived from the boundary values 𝑃𝑚1 ∗(0, 𝑐) = Λ × (1 + δ)𝑃𝑚0 ∗(𝑐) + 𝜒 ′∗(𝑐) (16) 𝑃01 ∗(0, 𝑐) = Λ × (1 + δ)𝑃00 ∗(𝑐) + 𝜒𝑃10 ∗(𝑐) (17) Theorem 1: For the Single server Markovian General service retrial encouraged arrival queuing system under the persistent retrial technique, a. The transitory result of the patrons in the group, when the system- server is free in the system, is provided by 𝑃0 ∗(𝑐, 𝑍) = 1 + 𝜒𝑃00 ∗ (𝑐) − 𝜒 𝑍 𝑃00 ∗ (𝑐)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (𝑐 + Λ × (1 + δ) + 𝜎) − (Λ × (1 + δ) + 𝜎 𝑍)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) b. The transitory result of the patrons in the group, when the system- server is engaged in the system, is provided by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 308 https://internationalpubls.com 𝑃1 ∗(𝑐, 𝑍) = (Λ × (1 + δ) + 𝜎 𝑍 )( 1 + 𝜒𝑃00 ∗ (𝑐) − 𝜒 𝑍 𝑃00 ∗ (𝑐)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (𝑐 + Λ × (1 + δ) + 𝜒) − (Λ × (1 + δ) + 𝜒 𝑧 ) 𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) ) − 𝜎 𝑧 𝑃00 ∗ 𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 (1 − 𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍)) c. The Steady-state result of patrons in the group when the server is free in the service system is provided by 𝑃0(𝑍) = 𝑝00 1 − ( Λ × (1 + δ) ∗ 𝑍 𝜒 ) 𝛿(𝑍) where 𝛿(𝑍) = 1 − 𝑎‾ 𝑎‾ − 𝑍 , 𝑃00 = 1 − 𝜌1 and 𝜌1 = Λ × (1 + δ)(Λ × (1 + δ) + 𝜒) 𝜒 𝐸(𝑌) d. The Steady-state result of patrons in the group when the server is engaged in the service system is provided by 𝑷𝟏(𝒁) = 𝑷𝟎𝟎( 𝜹(𝒁) 𝟏 − ( 𝚲 × (𝟏 + 𝛅)𝒁 𝝌 )𝜹(𝒁) ) Proof: We determine the probability-generating functions 𝜔(𝑦, 𝑇, 𝑍) = ∑   ∞ 𝑚=0  𝑃𝑚1(𝑦, 𝑇)𝑍 𝑚 (18) 𝑃0(𝑇, 𝑍) = ∑   ∞ 𝑚=0  𝑃𝑚0(𝑇)𝑍 𝑚 (19) Using the LT to the equations (18) and (19), we obtain 𝜔∗(𝑦, 𝑐, 𝑍) = ∑𝑚=0 ∞  𝑃𝑚1 ∗ (𝑦, 𝑐)𝑍𝑚 (20) 𝑃0 ∗(𝑐, 𝑍) = ∑𝑚=0 ∞  𝑃∗ 𝑚0(𝑐)𝑍 𝑚 (21) Partly differentiate (20) concerning 𝑦 𝜔′∗(𝑦, 𝑐, 𝑍) = ∑𝑚=0 ∞  𝑃′𝑚1 ∗ (𝑦, 𝑐)𝑍𝑚 (22) Substitute (12) and (13) in (20), we have 𝜔′∗(𝑦, 𝑐, 𝑍) + (𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 + 𝜇(𝑦))𝜔∗(𝑦, 𝑐, 𝑍) = 0 (23) When we solve the differential equation above, we obtain 𝜔∗(𝑦, 𝑐, 𝑍) = 𝜔∗(0, 𝑐, 𝑍)𝑒−(𝑐+Λ×(1+δ)−Λ×(1+δ)∗𝑍)𝑦𝑒−∫   𝑦 0  𝜇(𝑦)𝑑𝑦 (24) Where Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 309 https://internationalpubls.com 𝜔∗(0, 𝑐, 𝑍) = ∑   ∞ 𝑚=0  𝑃𝑚1 ∗ (0, 𝑐)𝑍𝑚 (25) Substituting (16) and (17) in (25), we obtain 𝜔∗(0, 𝑐, 𝑍) = (Λ × (1 + δ) + 𝜒 𝑍 )𝑃0 ∗(𝑐, 𝑍) − 𝜒 𝑍 𝑃00 ∗ (26) Multiply the equation (24) by 𝜇(𝑦) ∫ 𝑖𝑛𝑔 with respect x between 0 and ∞ ∫   ∞ 0  𝜔∗(𝑦, 𝑐, 𝑍)𝜇(𝑦)𝑑𝑦 = 𝜔∗(0, 𝑐, 𝑍)∫   ∞ 0   𝑒−(𝑐+Λ×(1+δ)−Λ×(1+δ)∗𝑍)𝑦𝑒−∫   𝑦 0  𝜇(𝑦)𝑑𝑦𝜇(𝑦)𝑑𝑦 (27) ∫   ∞ 0  𝜔∗(𝑦, 𝑐, 𝑍)𝜇(𝑦)𝑑𝑦 = 𝜔∗(0, 𝑐, 𝑍)∫   ∞ 0   𝑒−(𝑐+Λ×(1+δ)−Λ×(1+δ)∗𝑍)𝑦𝑎(𝑦)𝑑𝑦 (28) ∫   ∞ 0  𝜔∗(𝑦, 𝑐, 𝑍)𝜇(𝑦)𝑑𝑦 = 𝜔∗(0, 𝑐, 𝑍)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (29) where 𝑎‾(𝑐) = ∫ 0 ∞  𝑒−𝑐𝑦𝑎(𝑦)𝑑𝑦 is the LT of the service-system time distribution ∫ 𝑖𝑛𝑔 the equation (24) with respect to 𝑦 via 0 and ∞, we have 𝑃1 ∗(𝑐, 𝑍) = 𝜔∗(0, 𝑐, 𝑍) 𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 (1 − 𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍)) (30) Substitute the above equation (26) in (30), we obtain 𝑃1 ∗(𝑐, 𝑍) = (Λ × (1 + δ) + 𝜒 𝑍)𝑃0 ∗(𝑐, 𝑍) − 𝜒 𝑍 𝑃00 ∗ 𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 (1 − 𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍)) (31) Multiply the above equation (14) by 𝑍𝑚 on symmetry sides and add over 𝑚 = 1 to ∞, we obtain (𝑐 + Λ × (1 + δ) + 𝜒)∑1 ∞  𝑃𝑚0 ∗(𝑇)𝑍𝑚 = ∫ 0 ∞  (∑1 ∞  𝑃𝑚1 ∗(𝑦, 𝑐)𝑍𝑚)𝜇(𝑦)𝑑𝑦 (32) (𝑐 + Λ × (1 + δ) + 𝜒)(𝑃0 ∗(𝑐, 𝑍) − 𝑃00 ∗ (𝑐)) = ∫ 0 ∞  (𝜔∗(𝑦, 𝑐, 𝑍) − 𝑃01 ∗ (𝑦, 𝑐))𝜇(𝑦)𝑑𝑦 (33) Substitute the above equations (15) and (29) in (33), we obtain (𝑐 + Λ × (1 + δ) + 𝜒)𝑃0 ∗(𝑐, 𝑍) = 1 + 𝜎𝑃00 ∗ (𝑐) + 𝜔∗(0, 𝑐, 𝑍)𝑏‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (34) Substitute the above equation (26) in (34), we have (𝑐 + Λ × (1 + δ) + 𝜒)𝑃0 ∗(𝑐, 𝑍) = 1 + 𝜒𝑃00 ∗ (𝑐) + [(Λ × (1 + δ) + 𝜎 𝑍 )𝑃0 ∗(𝑐, 𝑍) − 𝜎 𝑍 𝑃00 ∗ (𝑐)] 𝑎‾ +Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (35) 𝑃0 ∗(𝑐, 𝑍) = 1 + 𝜒𝑃00 ∗ (𝑐) − 𝜒 𝑍 𝑃00 ∗ (𝑐)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (𝑐 + Λ × (1 + δ) + 𝜒) − (Λ × (1 + δ) + 𝜒 𝑍)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (36) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 310 https://internationalpubls.com In equation (36) means transitory results of the patrons in the group when the server is free in the system. 𝑃1 ∗(𝑐, 𝑍) = (Λ × (1 + δ) + 𝜎 𝑍 )( 1 + 𝜒𝑃00 ∗ (𝑐) − 𝜒 𝑍 𝑃00 ∗ (𝑐)𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (𝑐 + Λ × (1 + δ) + 𝜒) − (Λ × (1 + δ) + 𝜒 𝑍 ) 𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) ) − 𝜎 𝑍 𝑃00 ∗ 𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 (−𝑎‾(𝑐 + Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍)) (37) In equation (37) means transitory results of the patrons in the group when the server is engaged in the system. Using the Tauberian theorem with LT equation (35), we obtain (Λ × (1 + δ) + 𝜒)𝑃0(𝑍) = 𝜎𝑃00 + [(Λ × (1 + δ) + 𝜒 𝑍 )𝑃0(𝑍) − 𝜒 𝑍 𝑃00] 𝑎‾(Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) (38) Solving the equation (38), we obtain 𝑃0(𝑍) = 𝜒𝑝00(𝑍−𝑎‾) 𝑍(Λ×(1+δ)+𝜒)−(𝜒+Λ×(1+δ)𝑍)𝑎‾ (39) 𝑃0(𝑍) = 𝑝00 1−( Λ×(1+δ)𝑍 𝜒 )𝛿(𝑍) where 𝛿(𝑍) = 1−𝑎‾ 𝑎−𝑍̅̅ ̅̅ ̅̅ (40) In equation (40) means steady-state probability of patrons in the group when the system server is free. Using the Tauberian theorem with LT equation (31), we obtain 𝑃1(𝑍) = (Λ × (1 + δ) + 𝜒 𝑍)𝑃0 (𝑍) − 𝜒 𝑍 𝑃00 Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 (1 − 𝑎‾(Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍)) (41) Substitute the equation (40) in (41), we obtain 𝑃1(𝑍) = 𝑃00( 𝛿(𝑍) 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) ) (42) In equation (42) means steady-state probability of patrons in the group when the system server is free. The normalized state is 𝑃0(1) + 𝑃1(1) = 1 (43) 𝛿(1) = Λ×(1+δ)𝐸(𝑌) 1−Λ×(1+δ)𝐸(𝑌) = 𝜌 1−𝜌 where Λ×(1+δ) 𝜇 = Λ × (1 + δ)𝐸(𝑌) (44) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 311 https://internationalpubls.com 𝛿(1) = (Λ × (1 + δ))2𝐸(𝑌2) 2(1 − 𝜌)2 (45) 𝑃0(1) = lim𝑍→1  𝑃0(𝑍) = lim𝑍→1   𝑝00 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) = 𝑃00( 1 − Λ × (1 + δ)𝑍 𝜒 1 − ( Λ × (1 + δ)𝑍 𝜒 )1 ) (46) 𝑃1(1) = lim𝑧→1  𝑃1(𝑍) = lim𝑧→1  𝑃00( 𝛿(𝑍) 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) ) = 𝑃00( Λ × (1 + δ)𝑍 𝜒 1 − 𝜌1 ) (47) Substitute the equations (46) and (47) in (43), we obtain 𝑃00 = 1 − ( Λ×(1+δ)𝑍 𝜒 )1 (48) Theorem 2: The performance metrics of the Single server Markovian General encouraged arrival retrial queuing system under persistent retrial policy are provided below a. The probability of the system's server being free = 𝑃0 = 1 − Λ × (1 + δ) µ b. The probability of the system's server being engaged = 𝑃1 = Λ × (1 + δ) µ c. Mean number of patrons in the group L𝑞 = 1 (1 − ( Λ × (1 + δ) µ )1) ( (Λ × (1 + δ))2𝐸(𝑌2) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ) ∗ Λ × (1 + δ) µ 𝜒 ) d. Mean number of patrons in the system = 𝐿𝑠 = 𝐿𝑞 + Λ×(1+δ) µ Proof Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 312 https://internationalpubls.com When there is a steady state, equation (10) yields 𝑃0(𝑍) = ∑   ∞ 𝑚=0  𝑃𝑚0 ∗ 𝑧 𝑚 and 𝑃1(𝑍) = ∑   ∞ 𝑚=0  𝑃𝑚1 ∗ 𝑍 𝑚 (49) The probability of the server being free in the system = 𝑃0(1) = ∑   ∞ 𝑚=0  𝑃𝑚0 = 𝑃00( 1 − Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )1 ) = 1 − Λ × (1 + δ) µ (50) The probability of the server being engaged in the system = 𝑃1(1) = ∑   ∞ 𝑚=0  𝑃𝑚1 = 𝑃00 ∗ ( Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )1 ) = Λ × (1 + δ) µ (51) Differentiate the equation (40) for Z, we obtain 𝑃0 ′(𝑍) = 𝑃00 ( Λ × (1 + δ) 𝜒 ) ∗ (𝑍𝛿′(𝑍) + 𝛿(𝑍)) (1 − Λ × (1 + δ)𝑍 𝜒 𝛿(𝑍)) 2 (52) 𝑃0 ′(1) = 𝑃00 ( Λ × (1 + δ) 𝜒 ) ∗ (𝛿′(1) + 𝛿(1)) (1 − Λ × (1 + δ) 𝜒 𝛿(1)) 2 = ( Λ × (1 + δ) 𝜒 ) ∗ (𝛿′(1) + 𝛿(1)) ∗ (1 − Λ × (1 + δ) µ )2 (1 − Λ × (1 + δ) µ 1 ) (53) 𝑃0 ′(1) = ( Λ × (1 + δ) 𝜒 ) ∗ ( (Λ × (1 + δ)) 2 𝐸(𝑌2) 2 + Λ × (1 + δ) µ ∗ (1 − Λ × (1 + δ) µ )) (1 − Λ × (1 + δ) µ 1 ) (54) Differentiate the equation (42) via respect to Z, we obtain 𝑃1 ′(𝑍) = 𝑃0(𝑍) ∗ 𝛿 ′(𝑧) + 𝑃0 ′(𝑍) ∗ 𝛿(𝑍) (55) 𝑃1 ′(1) = ( Λ × (1 + δ)2𝐸(𝑌2) 2 + Λ × (1 + δ) 𝜎 ( Λ × (1 + δ) µ )2) (1 − ( Λ × (1 + δ) µ )1) (56) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 313 https://internationalpubls.com Mean number of patrons in the group = (𝑃′0(𝑍) + 𝑃1(𝑍)|𝑧=1 (𝑃′0(𝑍) + 𝑃1(𝑍))|𝑍=1 = ( Λ × (1 + δ) 𝜒 ) ∗ ( (Λ × (1 + δ)) 2 𝐸(𝑌2) 2 + Λ × (1 + δ) µ (1 − Λ × (1 + δ) µ )) (1 − ( Λ × (1 + δ) µ )1) + ( (Λ × (1 + δ))2𝐸(𝑌2) 2 + Λ × (1 + δ) 𝜒 ∗ ( Λ × (1 + δ) µ )2) (1 − 𝜌1) (57) = 1 (1 − ( Λ × (1 + δ) µ )1) ( (Λ × (1 + δ))2𝐸(𝑌2) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ) ∗ Λ × (1 + δ) µ 𝜒 ) L𝑞 = 1 (1 − ( Λ × (1 + δ) µ )1) ( (Λ × (1 + δ))2𝐸(𝑌2) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ) ∗ Λ × (1 + δ) µ 𝜒 ) (58) Average number of customers in the system = 𝑑 𝑑𝑍 (𝑃0(Z) + 𝑍𝑃1(𝑍))| 𝑍=1 = 𝐿𝑞 + 𝑃1(1) = 𝐿𝑞 + Λ × (1 + δ) µ 4. STABILITY CONDITION The consideration of stability has great significance for every queuing system. A stable single server retrial encouraged arrival queuing system is one that has a persistent retry policy. (1 + Λ × (1 + δ) 𝜒 ) ∗ Λ × (1 + δ)𝐸(𝑌) < 1 • If the stability condition of the exponential distribution is (1 + Λ × (1 + δ) 𝜒 ) Λ × (1 + δ) 𝜇 < 1 • If the stability condition of the Erlaang distribution is (1 + Λ × (1 + δ) 𝜒 ) 𝑟Λ × (1 + δ) 𝜇 < 1 5. SPECIAL PRIVILEGE We note that many particular cases of this work can be derived for various Service time distributions Privilege-1: An exponential distribution is used to describe the service time distribution. The PDF of exponential distribution are, 𝑎(𝑦) = 𝜇𝑒−𝜇𝑦, 𝑦 > 0 (59) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 314 https://internationalpubls.com E(Y) = 1 𝜇 and E(Y2) = 2 𝜇2 (60) The LT of 𝑎(𝑦) is 𝑎‾(𝑐) = 𝜇 𝑐+𝜇 (61) 𝑎‾(Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) = 𝜇 Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 + 𝜇 = 1 1 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍 (62) 𝛿(𝑍) = 1 − 𝑎‾ 𝑎‾ − 𝑍 = 1 − 1 1 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍 1 1 + ( Λ × (1 + δ) µ ) − ( Λ × (1 + δ) µ )𝑍 − 𝑍 = Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )𝑍 (63) 𝑃0(𝑍) = 𝑝00 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) = 𝑝00 1 − ( Λ × (1 + δ)𝑍 𝜒 )( Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )𝑍 ) (64) After the solution of equation (64), we obtain 𝑃0(𝑍) = 𝑃00(1− Λ×(1+δ) µ )𝑍) 1−( Λ×(1+δ) µ )1𝑍 where 𝜌 = Λ×(1+δ) 𝜇 and 𝜌1 = (Λ×(1+δ)+𝜒) 𝜒𝜇 (65) The equation (65) means the steady-state probability of patrons in the group, server is free in the system. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 315 https://internationalpubls.com 𝑃1(𝑍) = 𝑃00( 𝛿(𝑍) 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) ) = 𝑃00 ( Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )𝑍 1 − ( Λ × (1 + δ)𝑍 𝜒 )( Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )𝑍 ) ) (66) After the solution of equation (66), we obtain 𝑃1(𝑍) = 𝑃00( Λ × (1 + δ) µ 1 − ( Λ × (1 + δ) µ )1𝑍 ) (67) The equation (65) means the steady-state probability of patrons in the group, server is engaged in the system. L𝑞 = 𝑃00 (1 − ( Λ × (1 + δ) µ )1) 2( Λ × (1 + δ)2𝐸(𝑌2) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ) ∗ ( Λ × (1 + δ) µ ) 𝜒 ) = 𝑃00 (1 − ( Λ × (1 + δ) µ )1) 2( (Λ × (1 + δ))2 ( 2 𝜇2 ) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ)( Λ × (1 + δ) µ ) 𝜒 ) (68) After the solution of equation (68), we obtain L𝑞 = ( Λ × (1 + δ) µ )1 (1 − ( Λ × (1 + δ) µ )1) ( Λ × (1 + δ) µ + Λ × (1 + δ) Λ × (1 + δ) + 𝜒 ) (69) The equation (69) means the mean number of patrons in the group Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 316 https://internationalpubls.com L𝑠 = L𝑞 + Λ × (1 + δ) µ = ( Λ × (1 + δ) µ )1 (1 − ( Λ × (1 + δ) µ )1) ( Λ × (1 + δ) µ + Λ × (1 + δ) Λ × (1 + δ) + 𝜒 ) + Λ × (1 + δ) µ (70) The equation (70) means the mean number of patrons in the system. Case-2: An Erlang distribution under r phases describes the service time distribution. This privileges single server Erlang retrial queuing model under a persistent retrial technique The Erlang distribution is given that 𝑎(𝑦) = 𝜇𝑟𝑦𝑟−1𝑒−𝜇𝑦 (𝑟 − 1)! 𝜇𝑒−𝜇𝑦, 𝑦 > 0 For an Erlang distribution, E(Y) = 𝑟 𝜇 and E(Y2) = 𝑟(𝑟+1) 𝜇2 The Laplace transform of 𝑎(𝑦) is 𝑎‾(𝑐) = ( 𝜇 𝑐+𝜇 ) 𝑟 𝑎‾(Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) = ( 𝜇 Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍 + 𝜇 ) 𝑟 = ( 𝑟 𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍 ) 𝑟 (71) 𝛿(𝑍) = 1 − 𝑎‾ 𝑎‾ − 𝑍 = 1 − ( 𝑟 𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍 ) 𝑟 ( 𝑟 𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ ) ∗ 𝑍 ) 𝑟 − 𝑍 = (𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍)𝑟 − 𝑅𝑟 𝑅𝑟 − 𝑍 ∗ (𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍)𝑟 (72) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 317 https://internationalpubls.com 𝑃0(𝑍) = 𝑝00 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) = 𝑝00 1 − ( Λ × (1 + δ)𝑍 𝜒 )( (𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍)𝑟 − 𝑅𝑟 𝑅𝑟 − 𝑍(𝑟 + Λ × (1 + δ) µ − ( Λ × (1 + δ) µ )𝑍)𝑟 ) (73) Hereafter the solution of equation (73), we obtain 𝑃0(𝑍) = 𝑃00∗(1−𝑍∗(1+ 1 𝑅 ∗( Λ×(1+δ) µ −( Λ×(1+δ) µ )∗𝑍)) 𝑟 ) 1+𝑍( Λ×(1+δ) 𝜒 − 𝜌1 𝜌 (1+ 1 𝑟 (𝜌−𝜌𝑍)) 𝑟 ) where 𝜌 = Λ × (1 + δ) ∗ 𝐸(𝑌) and 𝜌1 = Λ×(1+δ)∗(Λ×(1+δ)+𝜒) 𝜒 ∗ 𝐸(𝑋) (74) The Eqn (74) means the number of patrons in the group, and the server is free. 𝑃1(𝑍) = 𝑃00( 𝛿(𝑍) 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) ) = 𝑃00( (𝑟 + 𝜌 − 𝜌𝑍)𝑟 − 𝑟𝑟 𝑟𝑟 − 𝑍(𝑟 + 𝜌 − 𝜌𝑍)𝑟 1 − ( Λ × (1 + δ)𝑍 𝜒 ) ( (𝑟 + 𝜌 − 𝜌𝑧)𝑘 − 𝑟𝑟 𝑟𝑟 − 𝑍(𝑟 + 𝜌 − 𝜌𝑍)𝑟 ) ) (75) Hereafter the solution of equation (75), we obtain 𝑃1(𝑍) = 𝑃00 ∗ ( (1 + 1 𝑟 ∗ (𝜌 − 𝜌 ∗ 𝑍)) 𝑟 − 1 1 + 𝑍 ∗ ( Λ × (1 + δ) 𝜒 − 𝜌1 𝜌 (1 + 1 𝑟 (𝜌 − 𝜌𝑍)) 𝑟 ) ) (76) The Eqn (76) means the number of patrons in the group, and the server is engaged. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 318 https://internationalpubls.com L𝑞 = 1 (1 − 𝜌1) ∗ ( (Λ × (1 + δ))2 ∗ 𝐸(𝑌2) 2 ∗ (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ) ∗ 𝜌 𝜒 ) = 1 (1 − 𝜌1) ( (Λ × (1 + δ))2 ( 𝑟(𝑟 + 1) 𝜇2 ) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ)𝜌 𝜒 ) (77) Hereafter the solution of equation (77), we obtain L𝑞 = 1 (1 − 𝜌1) ( 𝜆2 ( 𝑟(𝑟 + 1) 𝜇2 ) 2 (1 + Λ × (1 + δ) 𝜒 ) + 𝜆𝜌 𝜒 ) (78) The Eqn (78) means the number of patrons in the group. L𝑠 = L𝑞 + 𝜌 = 1 (1 − 𝜌1) ( (Λ × (1 + δ))2 ( 𝑟(𝑟 + 1) 𝜇2 ) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ)𝜌 𝜒 ) + 𝜌 (79) The Eqn (79) means the number of patrons in the system. Privilege-3: As 𝜒 → ∞, single server Markovian general service encouraged arrival queuing model The number of patrons in the system is provided by 𝑃(𝑍) = 𝑃0(𝑍) + 𝑍 ∗ 𝑃1(𝑍) = 𝑝00 1 − ( Λ × (1 + δ)𝑍 𝜒 ) ∗ 𝛿(𝑍) + 𝑍 ∗ 𝑃00( 𝛿(𝑍) 1 − ( Λ × (1 + δ)𝑍 𝜒 ) 𝛿(𝑍) ) (80) As 𝜒 → ∞, the equation (80), we have Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 319 https://internationalpubls.com 𝑃(𝑍) = 𝑃0(𝑍) + 𝑍 ∗ 𝑃1(𝑍) = 𝑃00 ∗ (1 + 𝑍𝛿(𝑍)) = 𝑃00 ∗ (1 + 𝑍 ( 1 − 𝑎‾ 𝑎‾ − 𝑍 ))𝑃00 ∗ (1 + 𝑍 ( 1 − 𝑎‾ 𝑎‾ − 𝑍 )) = 𝑃00 𝑎‾(1 − 𝑍) 𝑎‾ − 𝑍 𝑃00 = 1 − Λ × (1 + δ) ((Λ × (1 + δ)) + 𝜎) 𝜒 ∗ 𝐸(𝑌) = 1 − Λ × (1 + δ)𝐸(𝑌) (81) 𝑃(𝑍) = (1 − Λ × (1 + δ)𝐸(𝑌))(1 − 𝑍)𝑎‾(Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) 𝑎‾(Λ × (1 + δ) − Λ × (1 + δ) ∗ 𝑍) − 𝑍 (82) The Eqn (82) means the Pollaczek - Khinchine equation for single server Markovian general service encouraged arrival queuing model L𝑞 = 𝑃00 (1 − 𝜌1)2 ( (Λ × (1 + δ))2𝐸(𝑌2) 2 (1 + Λ × (1 + δ) 𝜒 ) + Λ × (1 + δ)𝜌 𝜒 ) = (Λ × (1 + δ))2𝐸(𝑌2) 2(1 − 𝜌) (83) The Eqn (83) means the mean number of patrons in the line. 6. NUMERICAL STUDY The parameters Λ × (1 + δ), 𝜇 and 𝜒 will be verified the stability condition. The performance metrics of this are expressed in tables for different service distributions. Λ δ Λ × (1 + δ) 𝜇 𝜒 5 0.1 5.5 10 10,30,50,70,…5000 Table 1: Performance metrics for Λ × (1 + δ) = 5.5 and 𝜇 = 10 for different parameters of 𝜒. 𝜒 P0 P1 Lq Ls Wq Ws 10 0.5 0.5 5.2297 5.7797 0.9508 1.0508 30 0.5 0.5 1.0174 1.8640 0.2389 0.3389 50 0.5 0.5 1.0174 1.5674 0.1851 0..2850 70 0.5 0.5 0.9083 1.4583 0.1651 0.2651 90 0.5 0.5 0.8516 1.4016 0.1548 0.2548 100 0.5 0.5 0.8324 1.3834 0.1513 0.2513 300 0.5 0.5 0.7232 1.2732 0.1315 0.2315 500 0.5 0.5 0.7025 1.2525 0.1277 0.2277 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 320 https://internationalpubls.com 700 0.5 0.5 0.6938 1.2438 0.1261 0.2261 900 0.5 0.5 0.6889 1.2389 0.1253 0.2253 1000 0.5 0.5 0.6873 1.2373 0.1250 0.2250 3000 0.5 0.5 0.6772 1.2272 0.1231 0.2231 5000 0.5 0.5 0.6752 1.2252 0.1228 0.2228 Remarks:1 Given an exponential distribution for the service distribution, Table 1 illustrates the effect of χ on the average number of Patrons in a group. Additionally, the following is implied: • The average number of patrons in the group reduces the retrial rate 𝜒 maximizes and then applies the encouraged arrival concept for this model to increase the arrival rates. This method using a single server encouraged arrival queuing system if 𝜒 is Maximum. Table 2: Performance metrics for Λ × (1 + δ), = 5.5, 𝜇 = 50 and 𝑟 = 3 for different points of 𝜒. 𝜒 P0 P1 Lq Ls Wq Ws 10 0.7 0.3 0.2086 0.3186 0.0379 0.0579 30 0.7 0.3 0.1220 0.2320 0.0222 0.0422 50 0.7 0.3 0.1056 0.2156 0.0192 0.0392 70 0.7 0.3 0.0987 0.2087 0.0179 0.0379 90 0.7 0.3 0.0948 0.2048 0.0172 0.0372 100 0.7 0.3 0.0935 0.2035 0.0170 0.0370 300 0.7 0.3 0.0855 0.1955 0.0156 0.0356 500 0.7 0.3 0.0839 0.1939 0.0153 0.0353 700 0.7 0.3 0.0833 0.1933 0.0151 0.0351 900 0.7 0.3 0.0829 0.1929 0.0151 0.0351 1000 0.7 0.3 0.0828 01928 0.0150 0.0350 3000 0.7 0.3 0.0820 0.1920 0.0149 0.0349 5000 0.7 0.3 0.0818 0.1918 0.0149 0.0.349 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 321 https://internationalpubls.com Table 3: Performance metrics for Λ × (1 + δ) = 8.8, 𝜇 = 50 and r = 3 for different points of 𝜒. 𝜒 P0 P1 Lq Ls Wq Ws 10 0.52 0.48 0.7537 0.9297 0.0856 0.1056 30 0.52 0.48 0.3781 0.5541 0.0430 0.0.630 50 0.52 0.48 0.3147 0.4907 0.0358 0.0558 70 0.52 0.48 0.2885 0.4645 0.0328 0.0528 90 0.52 0.48 0.2742 0.4502 0.0312 0.0512 100 0.52 0.48 0.2693 0.4453 0.0306 0.0506 300 0.52 0.48 0.2399 0.4159 0.0273 0.0473 500 0.52 0.48 0.2342 0.4102 0.0266 0.0466 700 0.52 0.48 0.2317 0.4077 0.0263 0.0463 900 0.52 0.48 0.2303 0.4063 0.0262 0.0462 1000 0.52 0.48 0.2298 0.4058 0.0261 0.0461 3000 0.52 0.48 0.2270 0.4030 0.0258 0.0458 5000 0.52 0.48 0.2264 0.4024 0.0257 0.0457 Remark:2 Given an Erlang distribution for the service distribution, Table 2&3 illustrates the effect of χ on the average number of Patrons in a group. Additionally, the following is implied: The average number of patrons in the group increases the retrial rate 𝜒 maximizes and then applies the encouraged arrival concept for this model to increase the arrival rates. This method using a single server encouraged arrival queuing system if 𝜒 is Maximum 7. ECONOMIC COST To optimize the system’s operating cost, we established a cost analysis.in single server Markovian encouraged arrival queuing system. We find out the following costs, • Waiting cost= (c) *(Wait time) • Operating cost= (R)*(server charge per time duration) • Estimated Total Cost (ETC)= Operating cost + Waiting cost Λ δ Λ × (1 + δ) 𝜇 r 𝜒 5 0.1 5.5 10 3 10,30,50,70,…5000 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 322 https://internationalpubls.com Table 4: 𝜒 Waiting time (Wq) Waiting Cost Operating Cost Total Cost 10 0.0379 0.3793 150 150.3793 30 0.0222 0.2217 150 150.2217 50 0.0192 0.1920 150 150.1920 70 0.0179 0.1794 150 150.1794 90 0.0172 0.1724 150 150.1724 100 0.0170 0.1700 150 150.1700 300 0.0156 0.1555 150 150.1555 500 0.0153 0.1526 150 150.1526 700 0.0151 0.1514 150 150.1514 900 0.0151 0.1507 150 150.1507 1000 0.0150 0.1505 150 150.1505 3000 0.0149 0.1490 150 150.1490 5000 0.0149 0.1487 150 150.1487 Table 5: Λ δ Λ × (1 + δ) 𝜇 r 𝜒 8 0.1 8.8 10 3 10,30,50,70,…5000 𝜒 Waiting time (Wq) Waiting Cost Operating Cost Total Cost 10 0.0856 0.8564 150 150.8564 30 0.0430 0.8463 150 150.8463 50 0.0358 0.4296 150 150.4296 70 0.0328 0.3576 150 150.3576 90 0.0312 0.3278 150 150.3278 100 0.0306 0.3116 150 150.3116 300 0.0273 0.3060 150 150.3060 500 0.0266 0.2727 150 150.2727 700 0.0263 0.2661 150 150.2661 900 0.0262 0.2633 150 150.2633 1000 0.0261 0.2617 150 150.2617 3000 0.0258 0.2612 150 150.2612 5000 0.0257 0.2579 150 150.2579 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 323 https://internationalpubls.com 8. CONCLUSION The single server Markovian general service retrial encouraged arrival queuing model under the persistent with retrial policy is calculated using the Exponential and Erlang distribution method. Hereafter we find out the number of patrons in the group, the server is a free/engaged concept of this model. This method using a single server encouraged arrival queuing system if 𝜒 value is Maximum. Additionally, we calculated the Economic cost analysis for this model. Applies the encouraged arrival concept for this model, more efficient results get it compare of the Poisson arrival model. Different special privileges are done by various service time distributions. Declaration: No conflict of interest References [1] Artalejo. J.R and G.I.Falin (2002). Standard and retrial queueing systems. A comparative analysis, Revista, Matematica, Complutense, 15 , pp 101-129 [2] Artalejo J.R and A. Gomez-Corral (2008). Retrial Queueing systems-a computational Approach, Springer. [3] Falin G.I (1990). A survey of retrial queues. Queueing Systems 7, No.2, pp 127-167. [4] Farahmand. K (1990). Single line queue with repeated demands, Queueing systems, 6, No.2, pp 223-228. [5] Kulkarni V.G (1983). On queueing systems with retrials. Journal of Applied Probability 20, No.2, pp 380-389. [6] Templeton J.G.C (1990). Retrial Queues, Queueing Systems 7, No.2, pp 125-227. [7] Yang T and Templeton J.G.C (1987). A Survey on retrial queues. Queueing Systems, 2, pp 201-233. [8] Choudhury, G.(2003). Some aspects of M/G/1 queueing system with optional second service: TOP, Vol.11, 141-150. [9] Choudhury, G. (2005). An M/G/1 Queueing System with Two Phase Service under D-Policy: Information and Management Sciences. Vol.16, No. 4, 1-17. [10] Artalejo,J.R. and Choudhury,G. 2004. Steady state analysis of an M/G/1 queue with repeated attempts and two-phase service: Quality Technology and Quantitative Management, Vol.1, 189-199. [11] Jehad Al-Jararha Kailash C. Madan 2003. An M/G/1 Queue with Second Optional Service with General Service Time Distribution: Information and Management Sciences, Vol.14, No.2, 47-56. [12] Madan, K.C., Abu-Dayyeh, W. and Saleh, M.F.2002. An M/G/1 queue with second optional service and Bernoulli schedule server vacations: Systems Science, Vol.28, 51-62. [13] Som BK, Seth S. An M/M/1/N queuing system with encouraged arrivals. Global Journal of Pure and Applied Mathematics. 2017; 13: 3443-3453. [14] Khan IE, Paramasivam R. Reduction in waiting time in an M/M/1/N encouraged arrival queue with feedback, balking and maintaining of reneged customers. Symmetry. 2022; [15] Khan IE, Paramasivam R, Performance Study of an M/M/1 Retrial Queueing System with Balking, Dissatisfied Customers, and Server Vacations , Contemporary Mathematics, Volume 4 Issue 3 (2023), 379-619 . https://ojs.wiserpub.com/index.php/CM/article/view/3117 https://ojs.wiserpub.com/index.php/CM/article/view/3117 https://ojs.wiserpub.com/index.php/CM/issue/view/cm.v4i32023.379-619