Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 441 https://internationalpubls.com Customer Impatience and Feedback Mechanism in MX/G/1 Retrial Queue with, Unreliable Server, Bernoulli Vacation and Customer Search Strategies K. Kaarunya*1, D. Sumitha2 *1Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore, Tamil Nadu – 601043, kaarunyakamaraj@gmail.com 2Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore, Tamil Nadu – 601043, drsumithad@gmail.com Article History: Received: 28-09-2024 Revised: 27-11-2024 Accepted: 08-12-2024 Abstract: This article examines a complex retrial queueing system characterized by unreliable batch arrival, three phases of service, customer impatience, feedback mechanism, Bernoulli vacation policy and customer search behaviors. In this system, customer arrivals follow a Poisson process. When the server is idle, one customer from the batch receives service while the others enter a retrial group. The server offers essential service to all the incoming customers. If the server is busy upon arrival, customers may choose to leave (balk) or wait in the retrial group for the server to become available. Customers who retry either rejoin the service if the server is free or abandon the system if their retrial attempts fail (renege). The server may randomly breakdown during any phases of service and the repair starts immediately. After completion of repair, the server continues to serve the interrupted customer in the system. After completing each phase of service, the customer may either opt for optional service, or join the retrial group as a feedback customer, or leave the system. Upon completion of the first essential phase or second optional phase, if the server is idle, it searches for customers if available in the retrial group with a certain probability. After completing the third optional phase, the server goes on a Bernoulli vacation with certain probability. The retrial, service, repair and vacation times are arbitrarily distributed. By using the supplementary variable technique, various system state measures and reliability measures are derived. The effects of various parameters on these system measures are analyzed through numerical examples. Keywords: Batch arrival, customer impatience, feedback, customer search and Bernoulli vacation. 1. Introduction In the past few decades, queueing models with retrials have significant contributions in the fields of computer networks and communication systems. The arriving customers are served according to first come, first served basis. In retrial queueing system, the customer who finds the server busy leaves the service area and joins the retrial group to get service after random amount of time. “Retrial queue” with phase services have been studied by various researchers. Senthil Kumar and Arumuganathan (2010) analyzed MX/G/1 retrial queue with two phase service and two types of repair. Zadeh (2015) investigated a batch arrival multi-phase queueing system with feedback and single vacation policy. Tomar and Shrivatsav (2020) obtained the transient and steady state analysis of unreliable three phase retrial queueing system. mailto:kaarunyakamaraj@gmail.com mailto:drsumithad@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 442 https://internationalpubls.com “Impatient customers” generally balk or renege at the system. Arrar et al. (2012) examined the behaviour of batch arrival retrial queue with impatient phenomenon. Sumitha and Udaya Chandrika (2016) modeled two phase retrial queue with impatient customers, Bernoulli vacation and orbital search. Arienzo et al. (2020) analysed group service of impatient customers in retrial queue. “Breakdowns” are unavoidable in many situations. If the server is experiencing issues, it needs to be repaired. Djellab (2002) presented a retrial queueing system with breakdown. Singh and Kaur (2017) performed the sensitivity analysis of unreliable server retrial queue with optional service, multi-phase repair and Bernoulli vacation. Kuki et al. (2020) suggested retrial queueing system with non-reliable server, collision and impatient customers. Tian et al. (2023) investigated M/M/1 retrial queueing model with breakdown, repair and setup times where the server will be closed down when the system is empty. After service completion, the unsatisfied customer joins the retrial group as a “feedback” customer. The feedback phenomena occurs in many retrial queueing systems. Shweta Upadhyaya (2014) discussed the retrial queue by considering both the batch arrival process and Bernoulli feedback. Pankaj Sharma (2018) studied retrial queueing system with feedback and modified vacation using generating function approach. Nila and Sumitha (2021) analyzed MX/G/1 retrial queueing system with priority, collision and feedback customers. “Vacation” generally means the time where the server may not be available due to maintenance. Wang (2012) discussed retrial queue with Bernoulli vacation. Radha et al. (2017) studied unreliable group arrival retrial queue with Bernoulli vacation. Single server constant retrial queue with blocked customers and Bernoulli vacation was analysed by Ke and Wang (2021). Mathavavisakan and Indhira (2023) provided an in-depth analysis, outlining key principles and relevant literature of retrial queueing system with Bernoulli vacations. “Search of customers” from the retrial group reduces the idle time of the server. Krishnamoorthy et al. (2005) investigated the behaviour of the M/G/1 retrial queueing system with nonpersistent customers and orbital search. Sumitha and Udaya Chandrika (2012) modelled a repairable M/G/1 retrial queue with vacation and orbital search and derived the reliability indices to predict the system behaviour. Murugan et al. (2019) examined the bulk arrival retrial queueing model with orbital search and exponentially distributed multiple working vacations. 2. Model Description • The unreliable batch arrival retrial queue with three phases of service, balking, reneging, feedback, Bernoulli vacation and search of customers are analysed. • The customers arrive at the system according to Poisson process with rate . Let J be the size of the arriving batch which is a random variable with k}kJ{P == , k=1,2,3… and )(k  be the probability generating function with first two moments 1c and 2c . • If the server is idle, then one of the customers in the batch gets the service and the rest of the incoming customers from the batch enters the retrial group. • Otherwise if the server is busy, then the arriving batch may join the retrial group with probability q or leaves the system with complementary probability. During retrials if the primary customer arrives in the system, the retrial customer cancels the attempt for service and return to Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 443 https://internationalpubls.com retrial group with probability p or renege at the system with complementary probability. The retrial time follows general distribution with distribution function ),(K  density function )(k  and Laplace Stieltjes Transform )s(K* and the hazard rate function ).(K1/)(k)( −= • The server provides three phases of service. The first essential service is provided to all the arriving customers whereas the second and third phases of service are optional. • After completion of first essential service, the customer leaves the system with probability 0r or opts for second optional service with probability 1 or moves to retrial group with probability )r1( 101 −−= for reservice. • Upon completing the second optional service, the customer leaves the system with probability 1r or proceeds to third optional service with probability 2 or moves to retrial group with probability )r1( 212 −−= for reservice. • After completing the third optional service the customer joins the retrial group with probability 3 or leaves the system with complementary probability. • Service times in all three phases are arbitrarily distributed with distribution function ),(Ji  density function ),(ji  Laplace Stieltjes Transform )s(J* i with first two moments 2i1i , and the hazard rate function .3,2,1i),(J1/)(j)( ii1i =−= • After the completion of first essential and second optional service, the server searches for customers in the retrial group with probability 2,1i,i = or remain idle with complementary probability. • At the completion of third optional service, the server may take a single vacation with probability  or waits for the next customer with complementary probability . Vacation times are arbitrarily distributed with distribution function ),(H  density function ),(h  Laplace Stieltjes Transform )s(H* with first two moments 21, and the hazard rate function ).(V1/)(v)( −= • In all the three phases of service the server is subjected to breakdown and the repair process starts instantly. The life time of the server in all the phases of service is exponential with rate .3,2,1i,i = The interrupted customer waits in the system to get service after repair completion. Repair times in all three phases were assumed to be arbitrarily distributed with distribution function ),(Fi  density function ),(fi  Laplace Stieltjes Transform )s(F* i with first two moments 2i1i , and the hazard rate function .3,2,1i),(F1/)(f)( iii =−= 2.1 System Analysis and Governing Equations The behaviour of the retrial queueing system at time t can be described by the Markov Process }0t),t(),t(),t(),t(),t(),t(),t(),t(),t(M),t(S{}0t),t(N{ 6543210 = At S(t)=0,1,2,3,4,5,6,7 represents the server states and M(t) represents the number of customers in the retrial group at time t. 0E is the probability that the server is idle at time t with no customers in the retrial group. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 444 https://internationalpubls.com ,d)t,(En  1n  denotes the probability that the server is idle at time t with n customers in the retrial group and the elapsed retrial time is between and .d+ ,d)t,(R n,i  ,0n  3,2,1i = denotes the probability that the server is busy in ith phase of service at time t with n customers in the retrial group and the elapsed service time is between and .d+  d)t,,(F n,i ,d ,0n  3,2,1i = the probability that the server is under repair in ith phase of service at time t with n customers in the retrial group and the elapsed repair time is between  and .d+ ,d)t,(Vn  ,0n  the probability that the server is on vacation at time t with n customers in the retrial group and the elapsed vacation time is between and .d+ The system of equations that governs the behaviour of the system is given below         +++= 0 0 0 0 030,3320,2110,100 d)()(Vd)()(Rd)()(Rrd)()(RrE (1) 1n),(E))(()(E nn +−=        (2)   +++−=        0 10,10,1110,1 d)(),(F)(R))(q()(R (3) 1n,d)(),(F)(Rq)(R))(q()(R 0 1n,1 n 1k kn,1kn,111n,1 ++++−=          = − (4)   +++−=        0 20,20,2220,2 d)(),(F)(R))(q()(R (5) 1n,d)(),(F)(Rq)(R))(q()(R 0 2n,2 n 1k kn,2kn,222n,2 ++++−=          = − (6)   +++−=        0 30,30,3330,3 d)(),(F)(R))(q()(R (7) 1n,d)(),(F)(Rq)(R))(q()(R 0 3n,3 n 1k kn,3kn,333n,3 ++++−=          = − (8) ),(F))(q(),(F 0,110,1 +−=        +   (9)  = − ++−=        +   n 1k kn,1kn,11n,1 1n),,(Fq),(F))(q(),(F (10) ),(F))(q(),(F 0,220,2 +−=        +   (11) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 445 https://internationalpubls.com  = − ++−=        +   n 1k kn,2kn,22n,2 1n),,(Fq),(F))(q(),(F (12) ),(F))(q(),(F 0,330,3 +−=        +   (13)  = − ++−=        +   n 1k kn,3kn,33n,3 1n),,(Fq),(F))(q(),(F (14) )(V))(q()(V 00 +−=        (15)  = − ++−=        n 1k knknn 1n),(Vq)(V))(q()(V (16) with boundary conditions    −   +++= 0 31n,33 0 0 3n,332n,2 0 211n,110n d)()(Rdx)()(Rd)()(Rrd)()(Rr)0(E 1n,d)()(Vd)()(Rd)()(R 0 n 0 21n,2 0 2211n,111 +++   −  − (17)     ++++= 0 20,222 0 0 10,111 0 11010,1 d)()(Rd)()(Rd)(Epd)()(EE)0(R   ++ 0 21,221 0 11,110 d)()(Rrd)()(Rr (18)     +− + =  +− = ++ ++++= 0 0 1n,111 0 2kn 1n 1k k 0 1kn n 1k k1n01nn,1 d)()(Rd)(Ep)(Epd)()(EE)0(R 1n,d)()(Rrd)()(Rrd)()(R 0 21n,221 0 11n,110 0 2n,222 +++   +  +  (19)   = 0 1n,11n,2 0n,d)()(W)0(R (20)   = 0 2n,22n,3 0n,d)()(R)0(R (21) 0n),(R)0,(F n,11n,1 = (22) 0n),(R)0,(F n,22n,2 = (23) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 446 https://internationalpubls.com 0n),(R)0,(F n,33n,3 = (24) 0n,d)()(R)0(V 3 0 n,3n =   (25) The normalizing condition is   =  =  =  =  =  ++++++ 0n 0 0 n,1 0n 0 n,3 0n 0 n,2 0n 0 n,1 1n 0 n0 dd),(Fd)(Rd)(Rd)(Rd)(EE    =  =  =  =++ 0n 0n 0 n 0 0 n,3 0n 0 0 n,2 1d)(Vdd),(Fdd),(F Define the probability generating functions   = = 1n n n )(E),(E ;   = = 0n n n,11 )(R),(R   = = 0n n n,22 )(R),(R ;   = = 0n n n,33 )(R),(R   = = 0n n n,11 ),(F),,(F ;   = = 0n n n,22 ),(F),,(F   = = 0n n n,33 ),(F),,(F ;   = = 0n n n )(V),(V Theorem 2.1 The partial probability generating function of the joint probability distribution for the server being idle, busy with three phases, under repair in three phases and on vacation are respectively given )(U/])(M)()(M))[(K1(E)(E 2 23 * 0 −+−= (26) ))(qq(G)(U/)))](qq(G(J1)][(M)([E)(R 11 * 1101 −−−−= (27) ))(qq(G)(U/)))](qq(G(J1))[(qq(G(J)](M)([E)(R 22 * 21 * 11012 −−−−−= (28) ))(qq(G)(U )))](qq(G(J1))[(qq(G(J))(qq(G(J)](M)([E )(R 3 3 * 32 * 21 * 11021 3 − −−−−− = (29) ))(qq/())](qq(F1)[(R)(F * 1111 −−−= (30) ))(qq/())](qq(F1)[(R)(F * 2222 −−−= (31) ))(qq/())](qq(F1)[(R)(F * 3333 −−−= (32) ))(qq/())](qq(V1)[(R)(V 3 −−−= (33) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 447 https://internationalpubls.com ]1NcqNc)[(K)NT))((K1(p Ncq)cp))((K1(T1 E 21312111 ** 21312111 * 0 −−−−−−++− −−−−−−+ = (34) where, )(M)(M)]pp)(())(K1()(K[)(U 32 **2 −+−+−= )pp)(())(K1()(K)(M ** 1 +−+= )()))(qq(G(J)r()r())){(qq(G(J)(M 2132132 * 221211011 * 12 ++−+++−= )))(qq(G(J)))(qq(G(J)))(qq(G(J)))(qq(G(J 2 * 21 * 1213 * 32 * 2 −−+−− ))}(qq(V)))(qq(G(J * 3 * 3 −− )))](qq(G(J)r()r()))[(qq(G(J)(M 2 * 221211011 * 13 −+++−= Proof To derive the probability generating function, we proceed by multiplying equations (2) to (25) by nz and summing over suitable powers of n, we get the resulting partial differential equations ))(K1(e),0(E),(E −= − (35) 3,2,1i,d)(),,(F),(R)(q)(q( d d 0 iiiii ==      −+++    (36) 3,2,1i)),(F1(e),,0(F),,(F i ))(1(q ii =−= −− (37) ))(V1(e),0(V),(V ))(1(q −= −− (38)       +++= 0 0 0 333 0 33322211110 d)(),(Rd)(),(Rd)(),(Rrd)(),(Rr),0(E       −+++ 0 0 0 0 22221111 Ed)(),(Vd)(),(Rzd)(),(Rz (39)        +   +   +  +   = 0 11 10 0 0 2 0 01 d)(),(R r d),(E )(p d),(E )(p d)(),(E 1 E )( ),0(R ++   +   d)(),(Rd)(),(Rd)(),(R r 2 0 2221 0 111 0 22 21 (40)   = 0 1112 d)(),(R),0(R (41) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 448 https://internationalpubls.com   = 0 2223 d)(),(R),0(R (42) 3,2,1i),,(R),0,(F iii == (43) =   d)(),(R),0(V 3 0 3 (44) Substituting equation (37) in (36) and solving, we get 3,2,1i)),(qq(F),0,(F),(R)(q)(q( d d 0 * iiiii =−=      −+++    (45) Substituting equation (43) in (37) and solving, we obtain 3,2,1i)),(F1(e),(R),,(F i ))(1(q iii =−= −− (46) Substituting equation (43) in (45) and solving, we get 3,2,1i)),x(J1(e),0(R),(R i ))(qq(Gi ii =−= −− (47) where, 3,2,1i)),(qq(F)(qq)(G * iiii =−−+−= Substituting equation (47) in (43) and solving, we get 3,2,1i)),x(J1(e),0(R),0,(F i ))(qq(Gi iii =−= −− (48) Substituting equation (47) in (44) and solving, we get )))(qq(G(J),0(R),0(V 3 * 33 −= (49) Substituting equation (47) in (41) and (42) and solving, we get )))(qq(G(J),0(R),0(R 1 * 1112 −= (50) )))(qq(G(J),0(R),0(R 2 * 2223 −= (51) Substituting equation (38) and (47) in (39) and solving, we get )))(qq(G(J)))(qq(G(Jrr)))[(qq(G(J),0(R),0(E 2 * 23212 * 2121101 * 11 −+−+−= ++−−+− 111113 * 32 * 23213 * 3 )))(qq(G(J)))(qq(G(J)))(qq(G(J 0 * 3 * 32 * 2212 * 2 E))(qq(V)))(qq(G(J)))(qq(G(J)))(qq(G(J −−−−+− (52) Using equation (35) and (47) in equation (40) and after some algebraic manipulations we get )(U/)](M)([E),0(R 101 −= (53) Substituting equation (53) in equations (50) and (52) and using the resulting equation in (51), we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 449 https://internationalpubls.com )(U/])(M)()(M[E),0(E 2 230 −+= (54) )))(qq(G(J),0(R),0(R 1 * 1112 −= (55) )))(qq(G(J)))(qq(G(J),0(R),0(R 2 * 21 * 11213 −−= (56) Using (53), (55) and (56) in (48) and solving, we get ))(J1(e),0(R),,0(F 1 ))(qq(G 111 1 −= −− (57) ))(J1(e),0(R),,0(F 2 ))(qq(G 222 2 −= −− (58) ))(J1(e),0(R),,0(F 3 ))(qq(G 333 3 −= −− (59) Substituting equation (56) in equation (49) and solving, we get )))(qq(G(J)))(qq(G(J)))(qq(G(J),0(R),0(V 3 * 32 * 21 * 1121 −−−= (60) Substituting expressions of equations ),0(V,),0(F,),0(F,),0(F,),0(R,),0(R,),0(R),,0(E 321321  from (53) to (60) in equations (35), (38), (46) and (47) we get the required results of equations (26) to (33). Using the normalizing equation and applying L’Hospital rule, 0E is obtained as in equation (34). 2.2 Performance Measures The probabilities that the server is idle, busy in all the three phases, under repair in all the three phases of service and on vacation are derived and given respectively as 121312111 * 0 U/]1NcqTc))[(K1(EE −++++−= (61) 111 * 1 * 01 U/)](Kcp))(K1[(ER +−= (62) 121 * 1 * 012 U/)](Kcp))(K1[(ER +−= (63) 131 * 1 * 0213 U/)](Kcp))(K1[(ER +−= (64) 11111 * 1 * 011 U/)](Kcp))(K1[(EF +−= (65) 12121 * 1 * 0122 U/)](Kcp))(K1[(EF +−= (66) 13131 * 1 * 02133 U/)](Kcp))(K1[(EF +−= (67) 11 * 1 * 021 U/)](Kcp))(K1[(EV +−= (68) where, 21312111 * 1 Ncq)cp))((K1(T1U −−−−−−+= 212121111T +−+−= 121313132121212111111 )1()1()1(N ++++++= Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 450 https://internationalpubls.com Theorem 2.2 The mean number of customers in the retrial group ( qL ) and in the system ( sL ) is given by )(Q d d ltL 1 q   = → 2 3 4132 S2 SSSS + = (69) 321321qs FFFRRRLL ++++++= (70) Proof )]}NT))((K1(p)(Kc[)]1c(TNcq)[(K{qcE2S ** 112131211 * 101 +−+−−++++= 21212111111111 * 11112 * 02 ()1(cq2)[(Kqc))1c(TNqc)(c2c(q)(K{E3S −+−++−++= )(k)()1(cq2) 2121211111222121221212112211 −+−++−++++ ]Hc2Tc22)1(cq)1(cq)1(2)(k 1212211212121111111212212122 ++++++−+−+ 3212121221212121111121 k)1(k)1(kH)c1(2T)cc2( ++−++−+−+++++ )1(2)1(cq)1(cq)1(2)cqcq( 21221212111111111122 2 1 22 21 −−++−+++ ))1(cq)1(cq()1(cq2)1(cq)1(cq 21212111111131313121212121111111 +++++++ 21321212121211211213111111 2)1(cq)()1(cq2 +++++−++−++ 11111112212121212313131212121 )1(cq2)1(cq)1(2))1(cq)1(cq( +++−++++ ))1(cq)1(cq)1(cq(cq2)1(cq( 3131312121211111111121212121 +++++++ 21112 ** 121 * 211 kkHqc2Tqc))](K1(p)(Kc[)NT)](cqc2))((K1()cc2[(qc ++++−+−++−−+− 21313131221212111111111121321 2))1(cq)1(cq()1(cq2cqk −++++−++ 21212111111111213131312212121 )1(cq)1(cq(cq2)1(cq)1(cq( +++−++ )})1(cq 313131 ++ )qc(U2S 113 −= )]qc(N)qc(U[3S 14214 −+−= Proof Let )(V)(F)(F)(F)(R)(R)(R)(EE)(Q 3213210 ++++++++= be the probability generating function of number of customers in the retrial group and )(R)(R)(R[)(EE)(H 3210 ++++= )(V)](F)(F)(F 321 ++++ be the probability generating function of number of customers in the system. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 451 https://internationalpubls.com )))(qq(G(J))))(qq(G(J1())(qq)((M))[(M)({(E)(Q 1 * 111 * 1210 −+−−+−−= )))(qq(G(J1)))((qq(G(J)))(qq(G(J))))(qq(G(J1( 3 * 32 * 21 * 1212 * 2 −−−−+−− )(K))](qq(V1)))((qq(G(J)))(qq(G(J)))(qq(G(J ** 3 * 32 * 21 * 121 −−−−−−+ ))(qq)((U/]})(M)()(M))[(qq( 23 −−+− (71) )))(qq(G(J))))(qq(G(J1(())(qq)((M))[(M)({(E)(H 1 * 111 * 1210 −+−−+−−= ))))(qq(G(J1)))((qq(G(J)))(qq(G(J))))(qq(G(J1( 3 * 32 * 21 * 1212 * 2 −−−−+−− )(K))](qq(V1)))((qq(G(J)))(qq(G(J)))(qq(G(J ** 3 * 32 * 21 * 121 −−−−−−+ ))(qq)((U/]})(M)()(M))[(qq( 23 −−+− (72) By differentiating the equations (71) and (72) with respect to Z and letting Z=1 we obtain, the mean number of customers in the retrial group ( qL ) and mean number of customers in the system ( SL ). Corollary 2.1 • Expected retrial group size when the server is idle in the non-empty system is derived as )](E[ d d limN 1 E   = → 2 3 4132 ]N[2 NNNN − = (73) where, ]1NcqTc))[(K1(EN 21312111 * 01 −++++−= )1()1(cq)(Ncq(c2)Tcc2)){((K1(EN 212111111212121111121 * 02 −−+−+−−++−= )Ncq(2))1(cq 121213213112211213121212121 +++++−−++++ 212123131311111121122 2 1 22 21321211 )1()1()1((cq2)cqcq(kkk ++++++++++ 2121211111121321212111111131313 )1()1((cq2))1()1((cq2))1( ++++++++ 2121211211111112111111131313 )1(cq2)1(cq2)1(cq2))1( ++++++++ }2]))1()1()1[(cq(cq2 31313212121111111121 −++++++ 21312111 * 3 Ncq)cp))((K1(T1N −−−−−−+= )(Ncq)]{pc))((K1()(K[2T)]pc2c))((K1[(2N 2121211111 ** 12 * 4 −+−−+−+++−−= })1(cq)1()1(cq 12211213121212121212111111 −−++++−−+ ))1()1((cq2)cqcq(kkk 212121111111122 2 1 22 21321211 ++++−−−− Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 452 https://internationalpubls.com 111111131313212123131311111121 )1(cq2))1()1()1()1((cq2 +−+++++− ))1()1((cq2))1()1()1((cq2 2121211111113131321212111111213 +++−+++++− 111111131313212121111111121 )1(cq2]))1()1()1[(cq(cq2 +−+++++− 122121211111121211111112111 2])1()1[(cq2)1(cq)(2 −+++++++− 12211212121 22)1(cq −−+ 12 2 111111112 2 1 22 112121 ]ccq[)]cqcq(qc[k ++++= 22 2 211212122 2 1 22 212222 ]ccq[)]cqcq(qc[k ++++= 32 2 311313132 2 1 22 312323 ]ccq[)]cqcq(qc[k ++++= • Expected retrial group size when the server is busy in first essential service is derived as )](R[ d d limN 1 1 R1   = → 2 7 8576 ]N[3 NNNN − = (74) where, 111111 * 1 * 05 )1(cq)](Kc)1p))((K1[(E2N +−−−= })1(cq)]pc2c))((K1()cc2[(k)](Kc)1p))((K1{[(E3N 11111112 * 211 * 1 * 06 ++−−+−−−−= ]1[cqU2N 111117 += ))]1(c(N))cac(cq(U[3N 1111412 2 1 22 1121218 +−++−−= • Expected retrial group size when the server is busy in second optional service is derived as )](R[ d d limN 2 1 R2   = → 2 11 1291110 ]N[3 NNNN − = (75) where, 212121 * 1 * 019 )1(cq)](Kc)1p))((K1[(E2N +−−−= ))1(cq)1(cq3k)]((Kc)1p))((K1{[(E3N 2121211111112 * 1 * 0110 +++−−−= })1(cq)]pc2c))((K1()cc2[( 21212112 * 21 ++−−+− ]1[cqU2N 2121111 += ))]1(c(N))cac(cq(U[3N 2121422 2 1 22 21222112 +−++−−= Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 453 https://internationalpubls.com • Expected retrial group size when the server is busy in third optional service is derived as )](R[ d d limN 3 1 R3   = → 2 15 16131514 ]N[3 NNNN − = (76) where, 313131 * 1 * 02113 )1(cq)](Kc)1p))((K1[(E2N +−−−= 3131311111113 * 1 * 02114 )1(cq)1(cq2k)]((Kc)1p))((K1{[(E3N +++−−−= })1(cq)]pc2c))((K1()cc2[())1(cq)1(cq2 31313112 * 21313131212121 ++−−+−+++ ]1[cqU2N 3131115 += ))]1(c(N))cac(cq(U[3N 3131432 2 1 22 31232116 +−++−−= • Expected retrial group size when the server is under repair in first essential service is derived as )](F[ d d limN 1 1 F1   = → 2 19 20171918 ]N[4 NNNN + = (77) where, }cq)1(cq)](Kc)1p))((K1{[(E6N 111111111 * 1 * 0117 +−−−= ))cqcq()1(cqcqk)]((Kc)1p))((K1{[(E12N 12 2 1 22 1121111111111 * 1 * 0118 +++−−−= )}cq)1(cq)](pc2c))((K1()cc2[( 11111111112 * 21 ++−−+− )]cq)(1(cqU[6N 11111119 −+= ))}1(c)(cq(N)]1(cq)cq()cq))(cac(cq[(U{12N 11111411112112 2 1 22 11212120 +−−++−+−+−−= • Expected retrial group size when the server is under repair in second optional service is derived as )](F[ d d limN 2 1 F2   = → 2 23 24212322 ]N[4 NNNN + = (78) where, }cq)1(cq)](Kc)1p))((K1{[(E6N 211212121 * 1 * 01221 +−−−= Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 454 https://internationalpubls.com )cqcq()1(cqcqk)]((Kc)1p))((K1{[(E12N 22 2 1 22 2122121212112 * 1 * 01222 +++−−−= )]pc2c))((K1()cc2[())cq)1(cq)1(cq(2 12 * 21211212121111111 +−−+−+++ )}cq)1(cq( 211212121 + )]cq)(1(cqU[6N 12121123 −+= )]1(cq)cq()cq))(cac(cq[(U{12N 21212122 2 1 22 21222124 +−+−+−−= ))}1(c)(cq(N 212114 +−−+ • Expected retrial group size when the server is under repair in third optional service is derived as )](F[ d d limN 3 1 F3   = → 2 27 25282726 ]N[4 NNNN + = (79) where, }cq)1(cq)](Kc)1p))((K1{[(E6N 311313131 * 1 * 021225 +−−−= )cqcq()1(cqcqk)]((Kc)1p))((K1{[(E12N 32 2 1 22 3123131313113 * 1 * 021226 +++−−−= )}cq)1(cq)](pc2c))((K1()cc2[( 31131313112 * 21 ++−−+− )]cq)(1(cqU[6N 13131127 −+= )cq(N)]1(cq)cq()cq))(cac(cq[(U{12N 1431312132 2 1 22 31232128 −++−+−+−−= ))}1(c( 3131 +− • Expected retrial group size when the server is on vacation is derived as )](V[ d d limN 1 V   = → 2 31 29323130 ]N[3 NNNN + = (80) where, }cq)](Kc)1p))((K1{[(E2N 11 * 1 * 02129 −−−= 11111111122 2 1 22* 1 * 02130 )1(cq(cq2)cqcq)](((Kc)1p))((K1{[(E3N +++−−−= }cq)]pc2c))((K1()cc2[()))1(cq)1(cq 1112 * 21313131212121 +−−+−++++ ))qc(U(2N 1131 −= )]c(N))qc(U[3N 141132 −+−= Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 455 https://internationalpubls.com 2.3 Reliability Measures Theorem 2.3 Availability of the server at time (t), is the probability that the server is idle or busy with customers is given by )](Kcp))(K1[()Ncq1)((K))(K1(Tp{EA * 1 * 2131211 ** 0 +−+−−−−+−= 1312121111 U/]}[ ++ (81) Proof The availability of the server can be expressed as )](R)(R)(R)(E[limEA 321 1 0 ++++= → (82) Substituting the equations (26) to (29) in (82) we get equation (81). Theorem 2.4 The failure frequency of the server is given by 1312132112111 * 1 * 0 U/)]((Kcp))(K1[(EM +++−= (83) Proof )](R)(R)(R[limM 332211 1 ++= → (84) Substituting the expressions in (26) to (29) in equation (84), we get (83). 3. Numerical Results The performance measures are illustrated numerically by using various system measures. For computation we choose the arbitrary parameters as ;2= ;7.0q = ;6.0p= ;6.01 = ;7.02 = ;8.03 = ;5.01 = ;6.02 = ;7.03 = ;5.01 = ;3.02 = ;8= ;9.0= ;2.0r0 = ;1.0r1 = ;3.01 = ;3.02 = ;251 = ;152 = ;103 = ;101 = ;82 = ;63 = ;10v= ;5.0c1 = .5.0c2 = The effects of various parameters on the system measures, −0E probability that the server is idle in the empty system, −E probability that the server is idle in the non-empty system, −R probability that the server is busy in service, −F probability that the server is under repair during service, −V probability that the server is on vacation, −sL the mean system size.    0E E R F V sL 0.6 3 10 0.7363 0.0229 0.2001 0.0173 0.0235 0.2921 15 0.7472 0.0215 0.1922 0.0156 0.0236 0.2828 20 0.7526 0.0208 0.1882 0.0147 0.0236 0.2781 6 10 0.7390 0.0226 0.2002 0.0148 0.0235 0.2897 15 0.7490 0.0212 0.1924 0.0138 0.0236 0.2812 20 0.7540 0.0206 0.1883 0.0134 0.0236 0.2769 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 456 https://internationalpubls.com 9 10 0.7399 0.0224 0.2003 0.0138 0.0235 0.2889 15 0.7496 0.0212 0.1924 0.0133 0.0236 0.2807 20 0.7545 0.0205 0.1883 0.0130 0.0237 0.2765 0.8 3 10 0.7345 0.0231 0.1999 0.0191 0.0235 0.2937 15 0.7460 0.0216 0.1921 0.0168 0.0236 0.2839 20 0.7517 0.0209 0.1881 0.0156 0.0236 0.2789 6 10 0.7381 0.0227 0.2001 0.0157 0.0235 0.2905 15 0.7484 0.0213 0.1923 0.0144 0.0236 0.2817 20 0.7536 0.0206 0.1883 0.0139 0.0236 0.2773 9 10 0.7393 0.0225 0.2003 0.0144 0.0235 0.2895 15 0.7492 0.0212 0.1924 0.0137 0.0236 0.2810 20 0.7542 0.0206 0.1883 0.0133 0.0237 0.2768 1 3 10 0.7327 0.0234 0.1998 0.0207 0.0234 0.2953 15 0.7448 0.0218 0.1921 0.0179 0.0236 0.2850 20 0.7508 0.0210 0.1880 0.0165 0.0236 0.2797 6 10 0.7372 0.0228 0.2001 0.0165 0.0235 0.2913 15 0.7478 0.0214 0.1922 0.0150 0.0236 0.2823 20 0.7531 0.0207 0.1882 0.0143 0.0236 0.2777 9 10 0.7387 0.0226 0.2002 0.0150 0.0235 0.2900 15 0.7488 0.0213 0.1923 0.0140 0.0236 0.2814 20 0.7539 0.0206 0.1883 0.0136 0.0236 0.2770 Table 1 System Measures by varying , and  Table 1 shows the effect of  ,, on the performance measures V,F,R,E,E0 and sL . It is clearly observed that • 0E increases with increase in  and  and decreases with increase in . • E and sL increases with increase in  and decreases with increase in  and . • R decreases with increase in  and  and increases with increase in . • F increases with increase in  and decreases with increase in  and . • V increases with increase in , decreases with increase in  and independent of . Influence of parameters  and  on E,E0 and sL are presented in Figures 1.1 to 1.3. 0E decreases with increase in  and increases with increase in . E increases with increase in  and decreases with increase in . sL increases for both the increasing values of  and . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 457 https://internationalpubls.com Figure 1.1 Effects of ),(  on 0E Figure 1.2 Effects of ),(  on E Figure 1.3 Effects of ),(  on sL The effect of parameters 1 and 1 on the system measures are given in Figures 1.4 to 1.6. 0E decreases with increase in 1 and increases with increase in .1 E increases with increase in 1 and decreases with increase in .1 sL increases with increase in 1 and decreases with increase in .1 Figure 1.4 Effects of ),( 11  on 0E Figure 1.5 Effects of ),( 11  on E Figure 1.6 Effects of ),( 11  on sL Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 4s (2025) 458 https://internationalpubls.com 4. Conclusion In this article, batch arrival retrial queueing system with essential and optional phases of service, customer impatience, breakdown, feedback mechanism, Bernoulli vacation and customer search behavior are discussed. Performance measures like the probability that the server is idle, busy, repair in all three phases of service and the server is on vacation are derived. The expected sizes retrial group and the system are derived. Reliability measures such as availability and failure frequency are also obtained. The present investigation can be further extended to cost optimization. References [1] Arrar, N.K., Djellab, N.V., & Baillon, J. (2012). On the asymptotic behaviour of M/G/1 retrial queues with batch arrivals and impatience phenomenon. Math. Comput. Model., 55, 654-665. [2] D’arienzo, M.P., Dudin, A.N., Dudin, S.A., & Manzo, R. (2020). Analysis of a retrial queue with group service of impatient customers. 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