TRANSIENT AND METAHEURISTIC COST SCRUTINY OF MX / G(A, B) / 1 RETRIAL QUEUE WITH RANDOM FAILURE UNDER EXTENDED BERNOULLI VACATION WITH IMPATIENT CUSTOMERS Текст научной статьи по специальности «Медицинские технологии»

TRANSIENT AND METAHEURISTIC COST SCRUTINY OF MX/ G(A, B)/ 1 RETRIAL QUEUE WITH RANDOM FAILURE UNDER EXTENDED BERNOULLI VACATION WITH IMPATIENT CUSTOMERS

Rani R1 and Indhira K *

^Vellore Institute of Technology , Vellore - 632 014, Tamil Nadu, India.

kindhira@vit.ac.in.

Abstract

The transient and metaheuristic cost analysis of a MX / G(a, b)/ 1 retrial queue with random failure during an extended Bernoulli vacation with impatient clients is covered in this study. Any batch that arrives and discovers the server is busy, down, or on vacation joins an orbit. In the alternative, only one new customer from the group joins the service right away, while the others join the orbit. After providing each service, the server either waits to serve the following customer with probability (1 — 8) or goes on vacation with probability 8. It has been found that these systems express steady-state solutions and are dependent on time probability generating functions in consideration of their Laplace transforms. We also discuss a few exceptional and particular instances. After that, the impact of different parameters on the system's effectiveness is evaluated. We are also talking about ANFIS. Additional approaches employed in this study to swiftly determine the system's optimum cost include genetic algorithms (GA), artificial bee colonies (ABC), and particle swarm optimization (PSO). We also examined the graph-based convergence of several optimization algorithms.

Keywords: Batch arrival, Retrial queues, Feedback, Extended Bernoulli Vacation, ANFIS, Cost Optimization.

1. Introduction

For the development, capacity planning, perfor mance assessment, and optimization of numerous real-world systems, queueing theory offers a potent tool. Chaudhr y and Templeton[ 1] provided a compr ehensiv e analysis of bulk queuing. Bulk arrival analysis, a condensed form of customer examination, is a great place to start with customised models. Bulk service queuing models were created by Bailey [2]. He invented the process known as "fixed-batc service". The server continuously offers a specifi batch of services to each set of users in fixed-batc service queueing systems (QS).

The "retrial queueing"system, which is used when a customer enters and the server is occupied, requir es the customer to lea ve the appr opriate area and repeat his request after a certain period of time. This property is essential for netw ork technologies, cognitiv e netw orks, online computing systems, manufacturing systems, and other systems.

Sumitha and Udaya Chandrika [3] investigated a retrial queuing system with starting failure, single vacation, and orbital search. In batch arrival retrial queues, Radha et al. [4] studied some system perfor mance measur es are evaluated using the supplementar y variable technique (SVT) and the steady-state (SS) probability generating function (PGF) for system size.

Gomez-Corral has talked a lot about a retrial QS with FCFS discipline and typical retrial

periods. The M/G/1 retrial queue with feedback and starting failures was described by Krishna Kumar et al. [5]. Yang, Tao, and Hui Li[6] investigated an M/G/1 retrial queue with a starting failure-prone server. An analysis of a feedbac k retrial queuing system with starting failures and a single vacation was studied by Mokaddis et al. [7].

In a Vacation, queueing system the server could be temporarily unavailable for a number of reasons, including maintenance monitoring, tending to other queues, or simply taking a break. When the server is unavailable to users, that time period is referred to as a "vacation ". A single server batch arrival Bernoulli feedback QS with awaiting server, K-variant vacations, and anxious clients was examined by Bouchentouf et al. [8]. The transient behaviour of a batch arrival feedback retrial queue with starting failure and Bernoulli vacation (BV) was investigated by Ayyappan and Sathiy a [9]. Assuming that repair, service, and vacation times are randomly distributed, the time-dependent PGF are also computed in relation to their Laplace transfor ms(LT).

Numer ous academics who have studied queueing techniques with interruptions have as their primary tenet that, in the event of a failure, the service channel will be promptly repaired. A transient analysis of the M[X1 MX]/ G1, G2/ 1 retrial QS's with priority services, working breakdown, start up/close down time, BV, reneging, and balking was studied by Ayyappan et al. [10]. Kulkar ni et al. [11] established a retrial queue with a server prone to failures and maintenance. Ayyappan and Shyamala [12] created an M[X1 / G/ 1 with Bernoulli schedule, server vacation, random break down and second optional repair. And also calculate the typical length of the line and the typical wait period in closed form. When the repair is finished a number of consumers who had previously used the services wait for the remainder to be provided. Jau-Chuan Ke et al. [13] demonstrated a waiting line with customers complaining and providing feedback the servers malfunctioned. Furthermore, if all servers are already in use when a customer arrives, he will either join a retrial orbit or decline. When a service is finished the client can exit the system or rejoin the retrial group to receive more services. They can also design a cost function to determine the system's ideal parameter settings under the stability condition. Computer telecommunication systems is a example of application for these types.

A consumer may try again until they are happy if they are not satisfie with the service they received. Takacs [14] investigates this at first allowing the consumer who has finishe the service to provide feedback to the rear of the line. An M/(G1, G2)/1 feedback retrial queue with two phase service, variant vacation policy under delaying repair for impatient Customers was analysed by Rajadurai et al. [15].

Many real-world systems have impatient custome rs as a built-in featur e, particularly when the customer is a human, a perishable product, or some moving object that can depart the service area and their waiting period in the queue reaches certain pre-define threshold values. This clearly explains why queueing literatur e frequently discusses the impatience phenomenon. Accounting impatience is crucial in the setting of lines for group service because a client could spend a lar ge amount of time in the system while waiting for the accumulation of a sufficien number of customers.

More focus has been placed on the numerous retrial lineups with non-persistent (impatient) consumers. A discussion about the study of a retrial queue wi th group service of impatient clients involved D'rienzo et al. [16]. A batch arrival retrial queuing model with starting failures and customer impatience was addressed by Nila and Sumitha [17]. Customers arrive in batches in line with the Poisson process. In certain situations, the clients refuse and break their promises. The analysis of a retrial QS with priority services, working breakdown, BV, admission control, and balking was explained by Ayyappan et al. [18]. Ayyappan and Nirmala [19] explored an analysis of customer 's impatience on bulk service QS's with an unreliable server, setup time and two types of multiple vacations. Sethi.R et al. [20] investigated the cost optimization and ANFIS computing of an unreliable M/M/1 queueing system with customers' impatience under n-policy. The ideal Cost Analysis for Discrete-Time Recurrent Queue with Bernoulli Feedback and Emergency Vacation was described by M. Vaishna wi [21]. In order to calculate costs, PSO, ABC, and GA are also used. To ensur e the best deal, these methods compar e and contrast the outputs.

The paper's structur e is as follows: Section 2 provides a detailed explanation of the mathematical model. Section 3 discusses the ideas and formulae governing our system as well as how to obtain the time-dependent solution of our model. The PGF for the queue length at each given epoch and the SS performance of the system are explicitly determined in Section 4. In Section 5, the pertinent stability condition has been uncovered. In Section 6, we precisely estimate the mean queue size, mean queue waiting time, and efficienc featur es for each state of the system. In Section 7, we present a practical illustration. We offer a numerical study and associated graphs in Section 8. Furthermore, an ANFIS was provided in Section 9. The Cost optimization is offered by Section 10. The conclusion is presented in Section 11.

2. Model Description and Analysis

We suppose that the underlying queueing model is as follows:

Arrival process: Customers enter a poisson stream, and bulk service is offered on an FCFS basis. Considering that a batch of "i"customers enters the system, A > 0 repr esents the average batch arrival rate, and Ac^dt(i > 1) represents the firs order probability during the short interval of time (©,© + d©]. We defin a batch arrival and a bulk service as having a smallest batch size of "a"and a highest batch size of "b".

Retrial process: When a customer arrives and discovers that the server is busy, unavailable, or broken, the customer has two options: (1) leave the service area with a probability of d and join a pool of blocked customers known as an orbit; or (2) balk the system with a probability of din accor dance with FCFS, which implies that only the customer at the head of the orbit queue is per mitted access to the ser ver.

When the server is idle, the customer at the head of the retrial queue engages with potential primary customers to see who can cancel their service request and, with prob., g, either move up in the retrial queue or leave the system with prob., (1 — g).

A general (arbitray) distribution with the distribution function A(u) and the density function a(u) deter mines the retrial inter val.

Let g(g)dg be the conditional prob., density of completing the retrial within the range (g, g + dg], wher e g is the elapsed retrial time.

g(g)- '{g)

1 — A(g)

and therefore,

a(u) = g(u)e— fo"g(g)dg

Inter-retrial times have an arbitrar y dist., A(g) with correponding Laplace-Stieltijes transforms (LST) A* (u).

Service process: The server enters an idle state wher ever a fresh or returning user comes before quickly resuming regular operations for the newcomers. A generic (arbitrary) distance with the distance function B(g) and the density function b(g) follo ws the ser vice time. Given the elapsed retrial time g, defin ty(g)dg as the conditional probability of service completion within the range (g,g + dg].

*)- b(g)

1 — B(g)

and therefore,

b(©) = <p(©)e— Joa<P(g)dg

The random variable B with the dist., function B(g) and LST B*(d) denotes the service time. Random failure: Failures are anticipated to occur sporadically throughout the system and ought to follow a poisson stream with an average failur e rate of t > 0. The repair times follow a general dist.„ which is represented by the random variable D and the dist., function D(g), with the LST D* (c).

The length of repairs is determined by a general (arbitrary) dist., with a dist., function D(g) and a density function d(g). Given an elapsed repair time of g, defin a(g)dg as the conditional probability of completing repairs within the range (g, g + dg].

a(g)- d(g)

1 - D(g)

and therefore,

d(c) = a(c)e— Jo a(g)dg

Extended Bernoulli vacation:If there are any unfinishe parts of the service, the server has two options: either accept the BV with a probability of 0 or keep serving them with a probability of (1 — 0). After the vacation is over, the server either undertakes the second type of optional extended Bernoulli vacation with a prob., of p or continues to serve the remaining batches with a prob., of (1 — p).

The random variable F with the distance function F(g) and LST F*(d) is employed to represent the ser ver's leisur e time. This arbitrar y variable F follo ws a general distribution. The server's vacation time follows a general(arbitrar y) dist., function F(d) and density function f(d). Let fi(g)dg be the conditional prob., of a completion of a vacation during the interval (g, g + dg], given that the elapsed repair time is g, so that

№)- f(g)

1 — F(g)

and therefore,

f (d) = p(o)e— J0^(g)dg

The system's stochastic processes are all consider ed to be independent of one another . Feedback Rule: Clients who are unhapp y with their offerings can re-join the line once they've been completed, give feedback to receive another service with minimal difficulty, or both p (0 < p < 1), otherwise the system must be terminated with complement prob. q = (1 — p)

3. DEFINITIONS:

We defin

1. Pn (g, d)= Prob., that the server will be idle at time D with n(n > 0) customers in the orbit and g for the customer 's elapsed retrial time.

2. Qn(g, d)= Prob., that the server will be busy at time D with n(n > 0) customers in the orbit and n for the customer 's elapsed retrial time.

3. Rn(g, d)= Prob., that at time D, there are n(n > 0) customers in the orbit and the server is offlin due to system repair and waiting for repairs to start with elapsed repair time g.

4. Vn(g, ©)= Prob., that there are n(n > 0) consumers in orbit at time © and the server is on vacation with elapsed vacation time g.

5. There are no customers in the orbit at time ©, and the server is inactive but still available in the system, according to the probability P0(©).

The following differential-dif ference equations regulate the model:

d I'M I'M

©0(©) = —AP0(©) + (1 — 6)dJo Q0(g, ©)$(g)dg + (1 — p) Jq V(g, ©)p(g)dg (1)

g g

—Pn(g,©) + g©Pn(g,©) = —[A + g(g)]Pn(g,©),n > 1 (2)

g g

— Q0 (g, ©) + g© Q0 (g, ©) = —[A + T + Q(g)]Q0 (g, ©) (3) g g n

—Qn(g,©) + g©Qn(g,©) = —[A + t + $(g)]Qn(g,©) + A£ CkQn—k(g,©),n > 1 (4)

g k=1

gg

—R0 (g, 1, ©) + g© R0 (g, ©) = —[A + a(g)]R0 (g, ©), n = 0 (5) g g n

—Rn(g, 1,©) + ^~Rn(g,©) = —[A + a(g)]Rn(g,©) + A£ CkRn—k(g,©),n > 1 (6) gg g© =

- V0 (g, ©) + g© V0 (g, ©) = —[A + p(g)]V0 (g, ©), n = 0 (7)

gg

n

- Vn (g, ©) + g© Vn (g, ©) = —[A + p(g)]Vn (g, ©) + A £ CkVn—k (g, ©), n > 1 (8) g k=1

The follo wing boundar y conditions must be met in order to answ er the given equation:

!• to !• to

Pn (0, ©)=(1 — 8)d Qn(g, ©)$(g)dg + (1 — 8)d Qn—1 (g, ©)$(g)dg

j0 j0

pTO /»TO

+ / Rn(g, ©)a(g)dg + (1 — p) Vn(g, ©)p(g)dg, n > 1 (9)

00

b a—1 .to

Q0(0, ©) =Ap(1 — g) £ £ Ck Pn—k+b(g, ©)dg

r=ak=0 j0

b to b , to

+ (1 — 8)p £ Pr(g, ©)g(g)dg + £ Vr(g, ©)p(g)dg (10)

r= a 0 r= a 0

a—1 p to p to

Qn(0, ©) =Ap(1 — g) £ Ck Pn—k+b(g, ©)dg + p Pn+b(g, ©)g(g)dg

k=0 0 - + 0 +

p to p to

+ Ag Pn+b (g, ©)dg +/ Vn+b (g, ©)p(g)dg (11)

00

R0 (g,0, ©)=t Q0 (g, ©), n = 0 (12)

Rn(g,0, ©)=tQn(g,©),n > 1 (13)

p to

Vn(0, ©) =8 Qn (g, ©)4>(g)dg, n > 1 (14)

0

We pr esume that the system is initially empty of users and that the ser ver is idle. Thus, the initial conditions are

Vn(0) = Rn(0) = Qn (0) = 0, n > 0

P0(0) = 1, P'n(0) = 0, n > 1 (15)

Generating functions of the queue length (The time-dependent solution):

œ œ

P(ç, Y, c) = £ Y^(ç, ©); P(Y,c) = £ Y"P„(©)

n=0 n=0

Q(ç, Y, c) = £ Y"Q„(ç, C); Q(Y, c) = £ Y"Q„(c)

n=0 n=0

œ œ

R(ç, i, Y, c) = £ Y"R„ (ç, I, CD); R(ç, Y, c) = £ Y"R„ (ç, c)

n=0 n=0

œœ

V(ç, Y, c) = £ Y"V„(ç, c); V(Y,c) = £ Y"V„(c)

n=0 n=0

œ a —1

C(Y) = E CnYn; Q(Y) = E QrYr (16)

n=1 r=0

which defin the LT of a function f (d) as it converges within the circle define by z < 1.

c

/(s) = e—sCDf (o)dcD, R(s) > 0 (17)

j0

Using (15) and the LT from equations (1) through (14), we arrive at

f f

(s + A)p0(s) = 1 + (1 — 0)dy Q0(g, s)^(g)dg + (1 — p)y V(g, s)^(g)dg (18)

d

—Pn(g, s) + [s + A + g(g)]Pn(g, s) = 0, n > 1 (19) d

— Q0 (g, s) + [s + A + ^(g)](30 (g, s) = 0 (20) d n

d- Qn (g, s) + [s + A + 0(g)]Q n (g, s) = AE Cfc^n—k (g, s), n > 1 (21) g k=1 d

dg R (g, i, s) + [s + A + a(g)]R0 (g, s) = 0 (22) d n

—Rn(g, i,s) + [s + A + a(g)]Rn(g,s) = AE CkRn—k(g,s),n > 1 (23)

g k=1

dg V0 (g, s) + [s + A + 0(g)] V0 (g, s) = 0 (24) d n

-V(g,s) + [s + A + 0(g)]Vn(g,s) = A E CkVn—k(g,s),n > 1 (25)

g k=1

/*M /*M

Pn (0, s) = (1 — 0)d Qn (g, s)^(g)dg + (1 — 0)d Qn—1 (g, s)0(g)dg

j0 j0

c m c M

+ / Rn(g,s)a(g)dg + (1 — p) Vn(g,s)0(g)dg, n > 1 (26)

00

b a—1 /.m

Q0 (0, s) = A p(1 — g) EE CW Pn—k+b (g, s)dg r=a k=0

b m b , m

+ (1 — 0)P E/ P(g,s)g(g)dg + E/ 7(g,s)0(g)dg (27)

r=a ^ 0 r=a ^ 0

a—1 m /• m

(3n(0,s) = Ap(1 — g) E CW Pn—k+b(g,s)dg + p/ Pn+b(g,s)g(g)dg

k=0 0 - 0

+ Ag / f'n+b(g, s)dg + / VVn+b(g, s)0(g)dg (28)

R0 (g,0, s) = t Q0 (g, s), n = 0 (29)

Rn(g,0, s) = tQn(g, s), n > 1 (30)

C m

Vn (0, s) = 0 Qn(g, s)0(g)dg, n > 1 (31)

By multiplying equations (19) through (31) by Yn and adding the results over n, we can obtain using the generating function mentioned in equation (16).

- P(g, Y, s) + [s + A + g(g)]P(g, Y, s) = 0 (32)

dg Q (g, Y, s) + [s + A(1 — C(Y)) + 0(g)]Q (g, Y, s) = 0 (33)

— R (g, i, Y, s) + [s + A(1 — C(Y)) + a(g)]R (g, Y, s) = 0 (34)

- V (g, Y, s) + [s + A(1 — C(Y)) + 0(g)] V (g, Y, s) = 0 (35)

C m f m

P(0, Y, s) = (1 — 0)(d"+ dY) ^ Q(g, Y, s)^(g)dg + j R (g, Y, s)a(g)dg

/»M _ /• M

+ (1 — p) V (g, Y, s)0gdg — d"(1 — 0) Q0 (g, s)^(g)dg J0 J0

/* m

— (1 — p)JQ I/0(g,s)0(g)dg, n > 1 (36)

/• M p M

YbQ(0,Y,s) = A(1 — g)pC(Y) / P(g,Y,s)dg + p P(g,Y,s)g(g)dg

/0 j0

f m i'm

+ Ag/ P(g, Y, s)dg +/ V (g, Y, s)0(g)dg (37)

00

J?(g, 0, Y, s) = tQ(g, Y, s), n > 1 (38)

C m

17(0, Y,s) = 0 Q(g, Y,s)0(g)dg, n > 1 (39)

0

Equation (18) in (36) gives us

f M

P(0, Y, s) = [1 — (s + A)P0(s)] + (1 — 0)(d" + dY) y Q(g, Y, s)^(g)dg

C m /* m

+ / R (g, Y, s)a(g)dg + (1 — p) 77 (g, Y, s)0(g)dg

00

Equation (32), when integrated between 0 and g, yields

P(g, Y, s) = P(0, Y, s)e—(s+A)g—S0 g(©)d© Once more, integrating equation (41) by parts with respect to g yiedls,

(41)

P(Y, s) = P(0, Y, s)

1 — A(s + A) s + A

wher e,

A(s + A) = r e—(s+A)gdA(g) 0

When integrating equations (33) to (35) from 0 to g, similar outcomes are found.

(42)

Q(g, Y, s) = Q(0, Y, s)e—z(Y,s)g—f° $(©)d© R(g, i, Y, s) = R(g, 0, Y, s)e—Z(Y,s)g—Jq a(©)d©

■ 1 — D (Z (Y, s))

R (g, Y, s) = R (g,0, Y, s)

Z (Y, s)

V(g, Y, s) = V(0, Y, s)e—ZzY)g—J0 ?(©)d©

where the values of P(0, Y, s),Q (0, Y, s),R (0, Y, s) and V (0, Y, s) are given by (37) to (40). Taking into account g yiedls, integrate equations (43) to (45) by parts once more.

Q (Y, s) = Q (0, Y, s) R (Y, s) = tQ (0, Y, s) V (Y, s) = V (0, Y, s)

1 — j?(Z (Y, s))

Z (Y, s) 1 — B(Z (Y, s)) Z (Y, s) 1 — F(Z (Y, s))

1 — D (Z (Y, s)) Z (Y, s)

Z (Y, s)

Wher e,

B(Z (Y, s)) D (Z (Y, s)) F(Z (Y, s))

' e—Z(Y,s)gdB(g) )

e—Z (Y,s)g dD(g)

)

e—Z (Y,s)g dF(g)

(43)

(44)

(45)

(46)

(47)

(48)

are, in order, the LST of the following values: retrial time A(g), service time B(g), repair time D(g), and vacation time F(g).

Now, multiplying both side of equations (41),(43) to (45) by g(g),ty(g),a(g) and fi(g) and integrating over g, we obtain

P(g, Y, s)g(g)dg = P(0, Y, s) A(s + A)

)

TO

Q (g, Y, s)$(g)dg = Q (0, Y, s)B(Z (Y, s))

)

R (g, i, Y, s)a(g)dg = R (g,0, Y, s)D (Z (Y, s))

TO

V (g, Y, s)p(g)dg = V (0, Y, s)F(Z (Y, s))

(49)

(50)

(51)

(52)

Using equations (50) in (39)

V (0, Y, s) = 0Q (0, Y, s)B(Z (Y, s)) Using equations (49) in (37) and (38), we get

P(0, Y, s)

Q (0, Y, s)

Yb - 0F(Z(Y,s))B(Z(Y,s))

A( 1 - g) PC(YW + pA(s + A) + Agi1 - A(s + A)

s + A

s + A

R(ç, 0, Y, s) = tQ(0, Y, s)

1 - B(Z(Y,s)) (Z (Y, s))

Using equation (50) to (52) in (40) we get

P(0, Y, s)

Nr(Y) Dr(Y)

(53)

(54)

(55)

(56)

Nr(Y) =[1 - (s + A)P0(s)][Yb - 0F(Z(Y,s))B(Z(Y,s))] Dr(Y) =Yb - 0F(Z(Y,s))B(Z(Y,s))

A(1 -g)PC(Y) () + pA(s + A) + Ag (l-l+lV

s + A

(1 - 0)(d + dY)B(Z(Y,s)) + tD(Z(y,s)) + 0(1 - p)F(Z(Y,s))B(Z(Y,s))

1 - B(Z(Y,s)) (Z (Y, s))

wher e,

Z (Y, s)= s + A(1 - C(Y))

Subs/- P(0, Y, s) from equation (56) into equation (53) to (55)

Q (0, Y, s)

A(1 - g)pC(Y) () + pA(s + A) + Ag ( 1:

A(s+a) s+a

Yb - 0F(Z(Y,s))B(Z(Y,s))

Nr(Y) Dr(Y)

RR (ç, 0, Y, s) =t(

1 - B(Z(Y,s)) (Z (Y, s))

Nr(Y) Dr(Y)

A(1 - g)pC(Y){ + pA(s + A) + Ag (i-

A(s+a) s+a

Yb - 0F(Z(Y,s))B(Z(Y,s))

V (0, Y, s) =0B(Z (Y, s))

Nr(Y) Dr (Y)

A(1 - g)pC(Y){ 1-+s+A)) + pA(s + A) + Ag (j-Yb - 0F(Z(Y,s))B(Z(Y,s))

A(s+a) s+a

(57)

(58)

(59)

Updating equations (56) to (59) in (42), (46) to (48) We determine the PGF of various conditions in the system under a transient condition.

4. The Steady state's findings:

To defin the SS prob., we disregard the argument D wher ever it appears in the time-dependent analysis.

lim s/?(s) = lim f (d)

wher e,

P(Y) = P(0, Y)( M^)

Q(Y)

1—™) P(0, Y)

A(1 — g)pC(Y) (+ pA(A) + Ag ^^

A(a) A

R(Y) = T (

Yb — 0 F(Z (Y))f?(Z (Y)) 1 — B(Z(Y))) (1 — D(Z(Y)))

z(y) ; v z(Y) )

P(0, Y)

A(1 — g)pC(Y) (+ pA(A) + Ag (^

Yb — 0 F(Z (Y))B(Z (Y))

V (Y) = 0 B(Z (Y))

1 — F(Z (Y)) \ Z (Y) J

P(0, Y)

A(1 — g)pC(Y) (+ pA(A) + Ag (^ Yb — 0 F(Z (Y))i?(Z (Y))

P(0, Y)

Nr(Y)

Dr(Y)

Nr(Y) = [1 — AP0 ] [Yb — 0 F(Z (Y))B?(Z (Y))] Dr(Y) = Yb — 0 F(Z (Y))B?(Z (Y))

A(1 — g) pC(Y)

1

A(A)\

A

+ pA(A) + Ag

1 — A(A)

A

(1 — 0)(d" + dY)B(Z (Y))+ t D (Z (Y)) +0 (1 — p)F(Z (Y))B(Z (Y))]

1 — B(Z (Y)) \ (Z (Y)) J

(60) (61)

(62)

(63)

(64)

(65)

4.1. Queue sizes distribution at a certain epoch:

The PGF is a of the queue size dist., at a random interval, is obtained by adding (60) to (63) with

the idle term.

K(Y) = №(Y)

K(Y) = Dr(Y) (66)

Nr(Y) = APqZ(Y) ( Yb - 9F(Z(Y))H(Z(Y)) - [(1 - g)pC(Y) (1 - A(A)) + pA(A)

1 - BB(Z(Y)) N

+g (1 - A(A))]

(1 - 9)(d"+ dY)H(Z(Y))+ tD(Z(Y))

(z(y)) ;

+9(1 - p)F(Z(Y))H(Z(Y))]) - (1 - A(A))Z(Y)[Yb - 9F(Z(Y))H(Z(Y))

+ A [(1 - g)pC(Y) (1 - A(A)) + pA(A) + g (1 - A(A))]

[(1 - HZ (Y)) + t (1 - HZ (Y))(1 - D Z (Y)) + 9 HZ (Y)(1 - FZ (Y))]]

+ (1 - A(A))Z(Y)[Yb - 9F(Z(Y))H(Z(Y))

+ A [(1 - g)pC(Y)(1 - A(A))+ pA(A) + g (1 - A(A))]]

[(1 - HZ (Y)) + t (1 - HZ (Y))(1 - D Z (Y)) + 9 HZ (Y)(1 - FZ (Y))]]

Dr(Y) = Z(Y)^Yb - 9F(Z(Y))H(Z(Y)) - [(1 - g)pC(Y) (1 - A(A)) + pA(A) + g(1 - A(A))] f(1 - 9)(d + dY)H(Z(Y)) + td(Z(Y)/1 - B(Z(Y))^

(z(y)) ;

+9(1 - p)F(Z (Y))H(Z (Y)))'

5. Stability Condition

The PGF needs to meet P(1)=1. Applying the L'Hopital rules and equating the expression to 1 results in the result that satisfie the requir ement.

b - [(1 -g)pE( I)(1 - A(A))][(1 - 9)(d + d) + 9(1 - p)] + p(1 - g)(1 - A(A)) + pA(A) + g(1 - A(A))[(1 - 9)(d + d)(1 - AE(I)E(H)) + tE(H) - 9A(1 - p)E(I)A1 ] + A9E(I)A1

Now we can determine the prob., that are unknown. P(1)=1 is therefore fulfille if

Yb - 9F(Z(Y))H(Z(Y)) - [(1 - g)pC(Y) (1 - A(A)) + pA(A) + g (1 - A(A))]

' 1 - H3(Z(Y)) -

> 0

(1 — 0)(d + dY)B(Z(Y)) + TD(Z(Y)) (1 (B(Y)Y)0 + 0(1 — ^)F(Z(Y))bB(z(Y))

[(1 — g) pE( I )(1 — A(A))] [(1 — 0 )(d + d) + 0(1 — p)] + [ p(1 — g)(1 — A(A)) +pA(A) + g((1 — aA(A))][(1 — 0)(d + d)(1 — AE( I )E(B))

_+ tE(B) — 0A(1 — p)E( I) A1 ] + A0 E( I) A1

p =-b- (67)

then p < 1 is the condition to be satisfie for the existence of the SS for the model under consideration.

6. Performance Evaluation:

This section includes system perfor mance metrics, a model stability study, and some unique system prob., while the system is in various states.

We obtain the following prob., if the system fulfill the stability requirement p < 1.

Let P be the SS Prob., that the server is idle during the retrial time.

P = lim P(Y) = P(1) = _(1 - 0)(1 - A Po)(1 -A(A))_=_

Y-mi ( ) () A(1 - 0) - [p(1 - g)(1 - A(A)) + pA(A) + g(1 - A(A))]

[(1 - 0)(d + d) + 0(1 - p)]

If the server is busy, let Q be the SS Prob.,

Q = fe Q(Y)

Q(1)=E(B)x

(1 - Apo)[p(1 - g)(1 - A(A))E(I)]

b + A0E(I)A1 - [(1 - g)pE(I)(1 - A(A))][(1 - 0)(d + d) + 0(1 - p)] + [p(1 -g)(1 - A(A)) + pA(A)+ g(1 - A(A))] (1 - 0)(d + d)(1 - AE(I)E(B)) + tE(B) - A0(1 - p)E(I)A1

R ought to indicate the SS Prob., that the server is being repaired.

R = lim R(Y)

-1

R(1) =t E(B)E(D)x

(1 - Apo)[p(1 - g)(1 - A(A))E(I)]

b + A0E(I)A1 - ([(1 -g)pE(I)(1 - A(A))][(1 - 0)(d + d)+ 0(1 - p)] + [p(1 -g)(1 - A(A)) + pA(A) + g(1 - A(A))] [(1 - 0)(d + d)(1 - AE(I)E(B)) + tE(B) - A0(1 - p)E(I)A1 ])

Using V as the SS Prob., we may assume that the server is on vacation.

V = lim V (Y)

-1

V(1) =0E(F)E(I) x

(1 - Apo)(-AE(B)p(1 - g)(1 - A(A)) + pA(A) + g(1 - Ai(A))

_+ p(1 - g)((1 - aA(A))))

b + A0E(I)A1 - ([(1 - g)pE(I)(1 - A(A))][(1 - 0)(d + d) + 0(1 - p)] + [p(1 - g)(1 - A(A)) + pA(A)+ g(1 - A(A))][(1 - 0)(d + d) (1 - AE(I)E(B)) + tE(B) - A0(1 - p)E(I)A1 ])

6.1. Average queue length:

Computing at Y = 1 and differentiating (65) with regard to Y yields the mean number of users in the queue (Lq) under SS conditions.

Lq = lim -d-P(Y) 7 y—1dY

P' (1)

Nr''(1)Dr'(1) - Dr''(1)Nr'(1) 2(Dr' (1))2

D (1) = -A2E(I) {Yb - 9F(Z(Y))B(Z(Y)) - [(1 - g)pC(Y) (1 - A(A)) + pA(A)

1 - B(Z(Y)) \

+g (1 - A(A))]

(1 - 9)(d + dY)B(Z(Y)) + tD(Z(Y))

(Z (Y)) ) +9(1 - p)F(Z(Y))B(Z(Y))]} D"(1) = -A2 {E(I(I - 1))[1 - 9 - [p -g(p - 1)(1 - A(A))][1 - 9p]]

+2E(I)[b + AdE(I)À1 - ([(1 - g)pE(I)(1 - A(A))] [(1 - d)(d + d) + 9(1 - p)] + [p(1 - g)(1 - A (A)) + pA(A)+ g(1 - A(A))][(1 - 9)(d + d)(1 - AE( I )E(B)) +tE(B) - A9(1 - p)E(I)À1 ])]}

N'(1) = - A2E(I) {1 - 9 - [p(1 -g)(1 - À(A)) + pA(A)+ g(1 - A(A))][(1 - 9)(d + d) +9(1 - p)]} + (1 - A) {-AE(I)(1 - 9)(1 - A(A)) + A2E(I)E(B) (p(1 -g)(1 - A(A)) + pA(A)+ g(1 - A(A))) + 9AE(I)E(F)} N''(1) = - A2 {E(I(I - 1)) (1 - 9 - A4(1 - 9p)) + 2E(I) (b + 9AE(I)A1 - A2(1 - 9p) +A4[(1 - 9) (1 - AE(I)E(B))] + tE(B) - 9A(1 - p)E(I)A1 )} + (1 - A) {(1 - A(A))[-(1 - 9)AE(I(I - 1)) - AE(I)(b + 9AE(I)A1 )] +A2E(I)E(B)A2 + A2[E(I(I - 1))E(B)+ E(I)E2(B)][p(1 -g)((1 - A(A))) +pA(A)+ g((1 - A(A)))] - 9A2E(B)E(F)E(I)2 +A9[E(I(I - 1))E(F) + E(I)E2(F)]}

wher e,

A1 =E(B) + E(F)

A2 =p(1 - g)E(I)((1 - Ai(A)))

A4 =p - g( p - 1)(1 - A (A))

The Little's formula (Wq) is used to determine how long an average customer waits in queue.

Lq

Wt- -q

q

AE(I)

7. Practical application of the model:

The fiel of telecommunications netw orks may be able to use the suggested model. This system manages a lot of consumer telephone communications. Call takers are referred to as servers and callers as customers in this context. A consumer may elect to exit the system if he calls and discovers that all the servers are occupied (impatience). Customers wait in orbit while the server is overloaded, out of commission, or undergoing maintenance. If a server has any questions or concerns that fall outside of their area of expertise, they may need to refer them to other ser vers who are available or speak with a senior in order to acquir e the answ ers. A service failure can be used to represent this circumstance. The speed at which the agent receives responses from the expert in this case is known as the repair rate. Additionally , the server may do various maintenance procedur es known as "vacations." Additionally , after each customer 's service is finished dissatisfie customers may re-join the line and be classifie as feedback consumers.

8. Numerical Results

In this section, we'll use MATLAB to demonstrate how different parameters affect observations of system behavior. The batch size distanc e of the arrivals in this section is geometr y; with a mean

of 2. Here, the exponential distance is followed by the service, vacation, and repair stages. By creating erroneous assumptions about the parameters, we make sure that the stability criterion is satisfied Tables 1 to 3 present estimated values for our queueing system's utilization factor (p), average queue length (Lq), and average waiting time (Wq).

Table 1: The effects of arrival rate (A) on p, Lq, and Wq

g = 0.5, p = 1.5, E = 0.6, G = 2.2, d = 3, d = 3, e = 0.6, p = 0.9, B = 1.5,D = 1, F = 0.7, z = 1,b = 2, t = 1.8

Arrival rate (A) p Lq Wq

0.30 0.022680 3.505127 5.841879

0.31 0.096336 4.616984 7.446748

0.32 0.169992 6.010404 9.391256

0.33 0.243648 7.742148 11.730527

0.34 0.317304 9.878047 14.526540

0.35 0.390960 12.494169 17.848813

0.36 0.464616 15.678118 21.775164

Table 2: The effects of the serv/ce rate ^(g) on p, Lq, Wq

g = 7.8, p = 0.7, E = 0.8, G = 6, d = 1, d = 3, e = 4.6, p = 0.7, D = 1, F = 0.7, A = 0.3, z = 1, b = 2, t = 1

service rate (B) p Lq Wq

0.50 0.737200 0.223375 0.372292

0.51 0.687360 0.189247 0.315412

0.52 0.637520 0.160488 0.267480

0.53 0.587680 0.135989 0.226649

0.54 0.537840 0.114943 0.191572

0.55 0.488000 0.096748 0.161247

0.56 0.438160 0.080947 0.134912

Table 3: The effects of the Breakdown rate (t) on p, Lq, Wq

g = 0.2, p = 0.7, E = 2.9, G = 9, d = 1, d = 7, e = 8.6, p = 0.2, B = 7,D = 2, F = 0.7, A = 0.4,z = 2, b

4

breakdo wn rate (t) p Lq Wq

1.0 0.264592 5.619523 7.024404

1.1 0.303092 6.095933 7.619916

1.2 0.341592 6.575234 8.219042

1.3 0.380092 7.057428 8.821784

1.4 0.418592 7.542516 9.428145

1.5 0.457092 8.030500 10.038125

1.6 0.495592 8.521381 10.651727

The two-dimensional graph that represents the system measur ement of performance is shown in Figur e 1 (a — c).

• The figu e 1 (a) demonstrates how the utilization factor (p), estimated queue length (Lq), and expected waiting time (Wq) all increase as the arrival rate (A) does.

• The figu e 1 (b) shows that while the utilization factor (p) decreases, the service rate <(g) rises. Expected waiting time (Wq) and queue length (Lq) decrease.

• The breakdown rate (t), utilization factor (p), expected queue size (Lq), and expected waiting time (Wq) all show increasing trends in the figu e 1 (c).

The three-dimensional graph of the system indicators of performance is shown in Figure 2 (a — c).

• The surface in figu e 2 (a) shows the growth of the arrival rate (A), estimated length of the line (Lq), and estimated wait time (Wq).

• Figure 2 (b) shows that as the service rate <(g) rises, the estimated queue size (Lq) and waiting time (Wq) both decrease.

• Figure 2 (c) shows that as the breakdown rate t rises, expected queue lengths (Lq) and waiting times (Wq) also rise.

The numerical results above allow us to determine the influenc of attributes on the system's evaluation criteria, and we can be assur ed that they are representativ e of realistic conditions.

(a) p, Lq, Wq verses arrival rate A

(b) p, Lq, Wq verses Service rate <(g)

(c) p, Lq, Wq verses Breakdown rate t

Figure 1: 2D representat/on effects

(a) p,Lq, Wq verses arrival rate A (b) p,Lq, Wq verses Service rate

<L,> 5 1 (c) p,Lq, Wq verses Breakdo wn rate t

Figure 2: 3D representation effects

9. Adaptive Neuro-Fuzzy Inference System (ANFIS)

The ANFIS modal is actually applicable in a variety of fields such as modes of transport, congestion, telecommuting, atmospheric research, etc. Artificia neural networks are used in communications networks to accomplish a variety of goals, including an increase in customers, expense reduction, shorter wait times, etc. With variations in arrival rates while on vacation, service rates, repair rates, and repair to busy rates, the current modal allows us to examine the impatience of the client while they wait for the service.

A very helpful appr oach for ANFIS is created by combining soft computing methods, artificia neural networks (ANNs), and fuzzy systems (FS). We are showing a simplifie idea of the ANFIS architectur e by using the fuzzy parameters. We can implement an ANFIS input-output function and input-output data pairs as fuzzy if-then logic. The fuzzy toolbox of MATLAB softw are can be utilized for contrasting the computational finding with the implementation of an ANFIS netw ork.

The input parameters and the membership function are assumed to be the A, $>(g), and t Gaussian functions in order to produce computational results based on ANFIS. (see Fig. 3a, b, c). It is assumed that the linguistic values are low, moderate, or high. Tick marks are placed over the curves made for the results obtained analytically in Figure 1a, 1b and 1c to indicate the results produced by the ANFIS approach for the queue size. The figu es show that the numerical outcomes produced using the Runge-Kutta method and the ANFIS results are nearly identical.

(a) p, Lq, Wq verses arrival rate A

(b) p, Lq, Wq verses Service rate $(g)

(c) p, Lq, Wq verses Breakdown rate T Figure 3: ANFIS representation effects

10. Cost Optimization:

The term "optimization" describes the method of deter mining the set of parameters for an objective function that produces the highest or lowest outcome. The continual, business-oriented activity known as "cost optimization" aims to reduce expenditur es and costs while raising the organization's value. Standar dizing, streamlining, and rationalizing platfor ms, application development, procedur es, and services are all part of this process, along with establishing the most competitiv e possible terms and prices for all business transactions. The operating cost and profi of a system are closely tied in real-w orld situations. Therefore, the system's designers or managers place a lot of emphasis on reducing operational expenses per unit of time in order to enhance the system's earnings. Our objective is to identify the best cost per unit of time (TC) characteristics. In order to do this and increase the cost-effectiveness of our developed approach, we will build our competence in this field

Ch - Holding expense for every user in the system per unit of time. Cb - The cost for each unit of time the server is turned on and used. Cv - The cost imposed on the server in vacation mode per unit of time. Cr - The cost to repair the ser ver after its failur e, calculated per unit of time. Ci - The cost per unit over a busy time.

C2 - Cost for each unit of time used over the vacation period.

TC = ChLq + CvV + CbQ + CrR + Ci Yb + C2 Yv

The TC problem is solved using metaheuristic optimisation methods including PSO, ABC, and GA. In view of the importance of cost optimisation, this study was conducted using the global search optimisation algorithms particle swarm optimisation (PSO), artificia bee colony (ABC), and genetic algorithms (GA), each of which is separately described in three different subsections of this section. If the algorithm's assumptions are correct, local search techniques frequently offer the level of computer efficienc requir ed to fin the global optimal. Tables 5 to 7 displa y the effects of A, t, and < on TC* using PSO, ABC, and GA.

Table 4: Cost sets for optimal policy

Cost sets Ch Cv Cb Cr Ci C2

1 10 9 7 6 7 8

2 8 4 6 4 8 9

3 7 6 8 3 9 6

10.1. Particle Swarm Optimization (PSO)

One of the meta-heuristic methods used to solve optimization issues is the particle swarm optimization (PSO) technique, which has been emplo yed successfully in a number of single objective optimization problems. Kennedy and Eberhart firs proposed this algorithm. The PSO algorithm has the benefi of being simple to implement and apply for solving different function optimization problems, which can be categorized as function minimization or maximization problems.

Table 5: Effect of A, t, p(ç) on TC* using PSO

g = 0.2, p = 0.7, G = 9, d = 0.95, d = 7, e = 8.6, c = 0.2, B = 7, D = 2, t = 1.6, b = 4

Cost sets TC*

Cost set 1 Cost set 2 Cost set 3

0.4 149.1752 133.0711 127.2781

A 0.5 162.6882 143.5244 136.3697

0.6 173.5857 152.1798 143.7058

1.6 149.1752 133.0711 127.2781

T 1.7 161.5959 141.8755 134.7960

1.8 175.6139 151.7985 143.2466

7 149.1752 133.0711 127.2781

№ 8 184.5033 159.0594 151.0219

9 230.2302 192.6975 181.7546

10.2. Artificial Bee Colony(ABC)

One of Dervis Karaboga's most recent algorithms—cr eated in 2005—is called the Artificia Bee Colony and was modeled after the cunning behaviour of honey bees. Basic process indicators like colonies and highest levels are essentially all that are used. Like PSO and differential evolutionar y appr oaches, it is equally simple to compr ehend. The sear ch for huge areas of nectar -containing

food sour ces, and ultimately the one with the most nectar, is the bees' mai n goal. This population-based sear ch appr oach is the main one used by ABC. The cost of the suggested structur e is decreased through a process known as ABC.

Table 6: Effect of A, t, p(g) on TC* using ABC

g = 0.2, p = 0.7, G = 9, d = 0.95, d = 7, e = 8.6, c = 0.2, A = 0.4, D = 2, t = 1.6, b = 4

Cost sets TC*

Cost set 1 Cost set 2 Cost set 3

0.4 108.0030 108.8585 109.6965

A 0.5 112.6458 113.7397 114.0666

0.6 115.8805 117.2631 116.9601

1.6 108.0030 108.8585 109.6965

t 1.7 112.4344 113.2737 114.1395

1.8 117.4656 117.8425 118.7674

7 108.0030 108.8585 109.6965

p(o) 8 120.7245 120.9982 122.5394

9 138.2173 134.5400 136.4184

10.3. Genetic Algorithm (GA)

The genetic algorithm, created in the 1960s and 1970s by Bremer mann, Holland, and their colleagues, is a technique for addr essing optimization problems brought on by natural selection, the mechanism that promotes evolution in biology. They are frequently employed to deliver superior solutions to stochastic sear ch issues. The full procedur e serves as a representation of the criteria for choice that were used to select the people who would make the best parents for the coming human generation.

Table 7: Effect of A, t, p(g) on TC* using GA

g = 0.2, p = 0.7, G = 9, d = 0.95, d = 7, e = 8.6, c = 0.2, A = 0.4, D = 2, B = 7, b = 4

Cost sets TC*

Cost set 1 Cost set 2 Cost set 3

0.4 152.5341 131.8364 124.6592

A 0.5 163.2414 141.3111 131.8853

0.6 170.0165 148.0781 136.4720

1.6 152.5341 131.8364 124.6592

t 1.7 167.5959 142.6723 133.7200

1.8 184.4744 154.7915 143.8279

7 152.5341 131.8364 124.6592

p(o) 8 192.8582 162.3320 151.7762

9 243.6750 200.7627 185.9493

10.4. Analogy of PSO, ABC and GA

This section compares the three approaches—particle swarm optimization (PSO), artificia bee colony (ABC), and genetic algorithm (GA)—to deter mine which has the least expense using the corresponding MATLAB programs. Then, one by one, the MATLAB programs for each of the aforementioned algorithms are run. We found that all three programs generated values that were nearly identical. Because of this, the three solutions are nearly comparable in terms of their optimum results and the fewest associated costs. It proves the reliability (local) and potency of these three simple techniques. Any technique can be used to calculate the optimal cost; however, PSO outperfor ms all others in comparison to our model. Because PSO has so many advantages, we have found that it is the best appr oach out of all of them. It perfor ms well in global queries, requires a small number of arguments, is easy to configu e, and is unaffected by design variable scalability. In addition to suffering sluggish convergence in a concentrated searching region, PSO has a tendency to lead to swift and early convergence in mid-optimal locations (being able to impair local sear ch capabilities).

10.5. Convergence in PSO, ABC and GA

After emplo ying an optimization methodology like PSO, ABC, or GA, it is crucial to compr ehend whether a particle recovers to normal or not and when it will roam around in search of a better solution. As a result, convergence is a significan component of cost evaluation. A statistical analysis (Fig. 4) of the outcomes demonstrates that ABC exceeds the PSO appr oach. For the whole standar d optimization, ABC had fewer functional evaluations overall than PSO. The finding demonstrate that PSO converges more quickly. ABC cannot be employed if a speedy result is requir ed for time-sensitiv e applications.

The study shows the applicability of our concept to real-w orld situations. Some of the analysts' financia issues will be partially overcome once they know how much the system will cost overall. The current situation may heavily rely on the cost-benefi assessment that was produced, which serves to illustrate the logic of our strategy and aid network administrators and specialists in lowering the issue of communications services that explicitly deal with blocking.

11. Conclusion:

This paper investigates the MX / G(a, b)/ 1 retrial queue with random failure and feedback under extended Bernoulli vacation with impatient customers. The SVT is utilized to determine indicators of efficiency for the various system stages. The efficienc of the system is then evaluated after considering the effects of various parameters. Finally, we gave a thorough explanation of the ANFIS. PSO, ABC, and GA are also used to compute the total cost. In an effort to fin the best offer, these techniques compar e and contrast the outcomes. The impetus for this study came from the prospective applications for the developed model, such as call centres, wireless networks, or telecommunication infrastructur es, which might be powered by controlled precision test queueing systems to provide outstanding service at low prices. The simple mail transfer protocol utilizes a w ay to conv ey the messages betw een the mail ser vers. The recommended appr oach might be used in an email system's transfer model.

Declarations:

Acknowledgments: Not applicable

Funding information: This research did not receive any specifi grant from funding agencies in

(a) TC verses Arrival rate (A) in PSO (b) TC verses Service pig) in ABC

0 10 20 30 40 so 60 70 m 90 100 Convergence in GA

(c) TC verses Breakdo wn rate (t) in GA Figure 4: Cost Optimization effects

the public, commercial, or not-for-profi sectors.

Conflicts of interest: The authors declar e no conflic of inter est.

Data availability: Not applicable

Authors contribution: All the authors made substantial contributions to the conception or design of the work.

Competing Interest: The authors declar e that they have no known competing financia interests or personal relationships that could have appear ed to influenc the work reported in this paper.

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