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UNRELIABLE M/G/1 QUEUE WITH GENERAL RETRIAL TIME, WORKING VACATION AND SETUP TIME
Hadjadj Houari , 1Arrar Nawel , 2Lahcene Yahiaoui
•
Probability and Statistics Laboratory, Badji Mokhtar University , BP12, Annaba, Algeria 2 Stochastic Models Statistics and Applications Laboratory, DTM Saida, Algeria 1*[email protected] ;1 [email protected] ; 2 [email protected]
Abstract
In the current article, a retrial queuing system with working vacations, interruptions, setup time, and perfect repair is analyzed. The scenario includes a server taking working vacations during empty periods without a complete halt of servicing customers; however, the rates of service remain reduced. Further, a setup time is included here, implying that if the server remains idle when a new customer enters, the state changes to inactive plus a setup duration before restarting operation. In this phase of setup, the setup failure happens and is replaced immediately before the server can proceed to normal operations. In addition to this, automatic power-off to conserve energy is there when no customer comes while the server is in vacation mode. Customers who find that the server cannot be accessed spend time waiting in retrial orbit instead of entering a normal queue. Here they're encouraged to try again for service after a random time. The steady state probability generating functions for system size and retrial group size are obtained by analyzing the system dynamics through the supplementary variable technique (SVT). Reliability and optimization analyses will be included in what will be studied from the system. Reliability concerns evaluating the chances of the server being available at different failure and repair sites while in the system, while optimization looks at the best configuration of system parameters that will work towards achieving greater efficiency and reduced delays. Explicit mathematical formulations can be obtained under ergodicity conditions describing the system size distribution and sojourn time and state probabilities. For a practical realization of the model, which numerically experiments would be carried out in Python, the theoretical results were validated .Such results therefore hold information on how direct retrials, setup times, service rates, and repair mechanisms affect overall system behavior. They also provide strong evidence for trade-offs between energy conservation on the one hand and reliability together with continuous service on the other. The proposed model together with practical implementation thus produces very significant inferences relevant to real service models in which the optimization of resources and efficiency of operation are critical.
Keywords: Retrial queue; working vacation; setup time; interruption; perfect repair ; general retrial times.
1. Introduction
Queueing theory is a fundamental branch of applied mathematics with widespread applications in various fields such as telecommunications, computer systems, transportation, and manufacturing. When servers are busy, customers often enter a retrial orbit, awaiting service availability according to predefined policies. The groundwork laid by Yang and Falin, Falin and Templeton [7], Yang et al. Extending by Artalejo, Gomez-Corral [8], Arrar et al [2] [3] [4]. Queueing systems, particularly those involving retrials with vacations play a crucial role in balancing resource utilization and service efficiency. The queueing systems with server vacations introduced by Levy and Yechiali [14] are widely used in manufacturing systems, production systems, service systems, inventory
systems, and other stochastic systems. In traditional scenarios, service stops when servers are on break, many researchers have worked on vacation interruption. Notably, Keilson and Servi [11] conducted significant research. One can refer also Li and Tian [15]. Takagi [21] contributed to the field by studying single-server queueing models with Bernoulli vacations. However, in many particular cases, an alternative approaches like working vacations provide service at a reduced rate during idle times, thereby enhancing the overall performance of the system. Numerous researchers have dedicated their efforts to develop models for queueing systems with working vacation concepts. Pioneering work on this subject was also carried out by Servi and Finn [18] considered an M/M/1 queue with working vacations, wherein vacation times are exponentially distributed. Most work on queueing models with working vacations can be found in Tian et al [19] and Chandrasekaran et al [6]. The M/M/1 retrial queue with working vacations was first studied by Do. Subsequently, Banik et al[5] analyzed a general working vacation queueing model. Arivudainambi et al [1] focused on analyzing a single-server retrial queue with working vacation dynamics. Zhang and Hou [20] studied M/G/1 queueing model with vacation interruption. And Gupta and Kumar [9] considered retrial queueing system with working vacation with breakdown and repair. Furthermore, the integration of setup time becomes essential, especially in the context of energy conservation. Setup time enables power-saving strategies by allowing server deactivation during periods of inactivity. Recognizing the importance of power conservation, several researchers have explored queueing models incorporating setup time. Phung-Duc [16] [17], for instance, integrated the notion of setup time into retrial queueing systems. Gupta and Kumar [10] analyzed retrial queue with feedback, setup time, working vacation perfect repair. For further insights into related research, readers may refer to Manoharan and Jeeva [12] [13]. And Pannom Gupta. The model under investigation is an M/G/1 retrial queue with a working vacation and setup time. This system represents a complex real-life case where customers may encounter delays due to server unavailability, requiring them to reattempt service after a waiting period. The setup times can significantly impact service efficiency and system performance. We extend the results obtained by considering the general law of inter-retrial times and service times. Our study aims to provide a detailed analysis of this type of complex system. We introduce the Markov chain to prove the stability condition of the studied model. Using the method of supplementary variables, we obtain the partial generating functions and the limiting distributions that this type of model can possess. We also present some performance measures of the studied model.
2. The model
In this paper, we consider an M/G/1 unreliable retrial queue with single working vacation , setup time. The model is described in great detail as follows:
1. The arrival of consumers follows a Poisson process with a rate A > 0 . If a consumer arrives and the server is idle, they begin service immediately . on the other hand, if he finds it busy, on working vacation, on setup, or broken then the consumer leaves the service area and enters a pool of blocked consumers called orbit in accordance with an First Come First Served discipline. While in orbit, the consumer waits a random amount of time before retrying. The inter-retrial times follow an arbitrary probability distribution function N(x) with corresponding LST function N* (9).
2. The server promptly initiates the normal service for the new or retrial consumers upon their arrival while in an idle state. The normal service time distributed with general distribution function G(x) having LST G*(9).
3. in the case of the orbit is empty.The server automatically begins a single working vacation, which follow an exponential distribution with rate w .If a consumers comes up during the working vacation , the consumers are served at a reduced rate. when the working vacation is finished the server resumes normal service . while the working vacation period , The service time distributed with general distribution function Wv(x) having LST Wj* (9).
4. In the final of working vacation , if no consumers are waiting for their turn , the server is shut down directly to save power .
5. in the off-state of the server, if any client arrives he will wait for his turn in forward position of the server until it is activated (setup time ). the setup time is assumed to follow general distribution with probability distribution function Ts (x) with corresponding LST function Ts* (9). The customers arriving in the setup state, have to join the orbit.
5. whene the server undergoes to the set-up state , this operation may fail with probability a = (1 — a). Then the server is sent for repair and The repair time of the server has arbitrary probability distribution function Sf (x) with corresponding LST function Sj (9).
2.1. Practical justifications of the suggested model
Consider a manufacturing system consisting of a paper recycling machine , a Foreman(server) and a worker(assistant) to operate the machine . The foreman will operate the machine if the waste paper (customer) is available and produce the products.if this last is not available due to transport issues , then the foreman may go on vacation .During the vacation period of the foreman, if waste paper becomes available then the worker will operate the machine , but the production will be relatively at a slow speed (working vacation ) . When a batch of the product is completed, then the worker will call the foreman to resume the production at a higher speed (vacation interruption ). In another situation, if the foreman's vacation period completes, he will return to the production to operate the machine . If the waste paper is available then he will manage the production at a higher speed otherwise, if the waste paper is not available, to save power, he may turn off the machine. Again, the availability of new waste paper will initiate the setup of the machine (setup time) and production starts again if setup occurs successfully otherwise the machine will be sent for repair, and during this period there will be no production.
3. Steady-state analysis
Let Xi(t),X2(t),X3(t),X4(t),X5(t) be the elapsed time in retrial, time in regular service, working vacation time , repair time and setup time sequentially at time t. Let supose that:
N (0) = 0, N(x) = 1, G(0) = 0, G(x) = 1, Wv (0) = 0, Wv (x) = 1, Ts (0) = 0, Ts (x) = 1, Sf (0) = 0, Sf (x) = 1 are continuous at x = 0.consequently, we specify the hazard rate functions f (x), ^n (x), ^w(x), v(x), 5(x), for retrial, normal service, lower rate service, delayed repair and repair, respectively.
f( x) dx ^(x)dx (x)dx v(x)dx 5(x)dx
dN(x) 1 — N (x) dG(x) 1 — G(x) dWv (x) 1 — Wv(x)
dSf(x) 1—f) dTs(x)
1 — Ts (x)
The state of the system at time t can be defined by the Markov process {N(t); t > 0} = {D(t),X(t),X1 (t),X2(t),X3(t),X4(t),X5(t)t > 0}, where X(t) denotes the number of customers in the orbit at time t and
D(t)=
0, if the server is idle in a normal period
1, if the server is idle in a working vacation period
2, if the server is busy on normal service
3, if the server is busy on working vacation period
4, if the server is on repair
5, if the server is on setup state
respectively. If D(t) = 0 and X(t) > 0, then X1 (t) represents the elapsed retrial time, if D(t) = 2, then (t) represents the elapsed service time during normal busy period at time t, if D(t) = 3 and X(t) > 0 then x3(t) represents the elapsed working vacation time at time t and if D(t) = 4 and X(t) > 1 then x4(t) represents the elapsed repair time at time t and if D(t) = 5 and X(t) > 1 then x5(t) represents the elapsed setup time at time t.
3.1. Stability and ergodicity Condition
Let {tn; n € N} be the sequence of epochs of either service completion times or vacation termination time. The sequence of random vectors Zn = {D (tn + ), X (tn + )} form a Markov chain which is the embedded Markov chain for our queueing system. Its state space is S = {0,1,2,3,4 and
5} x N.
Theorem 1. The embedded Markov chain {Zn; n € N} is ergodic if and only if
p = + + uw*'(u)) < N(A).
3.2. Equations Governing The System
For the Markov process {N(t); t > 0}, we define the probability
P00 (t) = {D(t) = 0, X(t) = 0}Q0(t) = {D(t) = 1, X(t) = 0} and the probability densities
Pn (x, t)dx = {D(t) = 0, X(t) = n, x < X1 (t) < x+dx},x > 0, n > 1
Mn,b (x, t)dx = {D(t) = 2, X(t) = n, x < X2(t) < x + dx},x > 0, n > 0
Gn,v (x, t)dx = {D(t) = 3, X(t) = n, x < X3(t) < x + dx},x > 0, n > 0
Un (x, t)dx = {D(t) = 4, X(t) = n, x < X4(t) < x + dx},x > 0, n > 1
Kn (x, t)dx = {D(t) = 5, X(t) = n, x < X5(t) < x + dx},x > 0, n > 1
We assume that the stability condition is fulfilled in the sequel and so that we can set
P00 = limt^w P00(t) and Q0 = limt^w Q0(t) limiting densities for x > 0 and n > 0
Mnb (x) = lim Mn b (x, t), Pn (x) = lim Pn (x, t) and Gn,v (x) = lim Gn,v (x, t) and Un (x) = lim Un (x, t) and Kn (x) = lim Kn (x, t).
t^w t^w
Based on the above assumptions and notations, our model is governed by the following set of differential difference equations,
AP00 = u Q0 (1)
w w
(A + u)Q0 = M0,b(x)^n(x)dx W G0,v(x)^w(x)dx (2)
00
ddxPn(x) + (A + f (x))Pn(x) = 0, x > 0, n > 1 (3)
dxM0,b(x) + (A + Vn(x)) Mo/b(x) = 0, x > 0 (4)
dj-M-nb(x) + (A + Vn(x)) Mn,b(x) = AMn-1,b(x), x > 0, n > 1 (5)
dJC
dxG0,v(x) + (A + w + Vw(x)) Gov(x) = 0, x > 0 (6)
dxGn,v(x) + (A + w + Vw(x)) Gn,v(x) = AGn-i,v(x), x > 0, n > 1 (7)
jxU0(x) + (A + v(x))U0(x)= 0, x > 0 (8)
dJC
j-Un(x) + (A + v(x))Un(x) = AUn-1(x), x > 0, n > 1 (9)
dJC
jxK0(x) + (A + S(x))K0(x)= 0, x > 0 (10)
j-Kn(x) + (A + S(x))Kn(x) = AKn-1(x), x > 0, n > 1 (11)
dJC
The boundary conditions at x = 0 include Pn (0), M0,b (0), Mn,b (0), G0,v (0), U0 (0), Un (0), K0 (0). The description of these terms at x = 0 are described as follows:
TO TO
Pn (0) = Gn,v (x)vw(x)jx + I Mn,b (x)vn (x)dx, n > 1 (12)
J0 J0
TO TO
M0,b(0) = ay K0(x)5(x)dx ^ Pi(x)f (x)dx
TO TO
+ w / G0v(x)dx + / U0(x)v(x)dx (13)
J0 ' J0
TO TO TO
Mn b(0) = a Kn (x)ô(x)dx + Pn+1 (x) f (x)dx + w Gnv(x)dx
J0 J0 J0
TO TO
+ A J Pn(x)dx + J Un(x)v(x)dx, n > 1 (14)
G0,v (0) = AQ0 (15)
Gn,v (0)= 0, n > 1 (16)
K0 (0) = AP00 (17)
Kn (0) = 0, n > 1 (18)
(• TO
U0 (0) = a K0(x)S(x)dx (19)
J0
TO
Un(0) = âj^ Kn(x)S(x)dx, n > 1 (20)
The normalization condition is given by
TO TO TO TO TO TO
P00 + Q0 + ^ Pn (x)dx + Y Mn,b (x)dx + Y Gnv (x)dx
n=0j0 n=0j0
TO TO TO TO
+ Y Un (x)dx + Y Kn (x)dx = 1 = J0 J0
n=0j0 n=0'
In order to solve the above set of equations we define the generating functions as, P(x, z) = EX=1 znPn(x) for |z| < 1 and x > 0,P(0,z) = EX=1 znPn(0) for |z| < 1, Mb(x,z) = EX=0 znM„ib(x) for Izl < 1 and x > 0, Mb(0, z) = EX=0 znMnb(0), Gv(x, z) = EX=0 znGn,v(x) for |z| < 1 and x > 0, Gv(0,z) = EX=0 znGn,v(0), U(x,z) = EX=0 znUn(x) for |z| < 1 and U(0,z) = EX=0 znUn(0) for z < 1 , K(x, z) = Ex=0 znKn (x) for z < 1 and K(0, z) = ^x=0 znKn (0) for z < 1. Multiplying equations (2)-(11) by suitable powers of z and summing over n, we obtain the following set of partial differential equations :
+ (* + f (x))P(x, z)= 0, (21)
dMb(x,Z) + (A - Az + ^(x)) Mb(x,z) = 0, (22)
Ox
dGv (x, z)
dx dU(x, z)
dx dK(x, z) dx
Solving the above partial differential equations (22) to (26)
P(x,z) = P(0,z)[1 - N(x)]e-Ax, (26)
Mb(x,z) = Mb(0,z) [1 - G(x)] e-I(z)x, (27)
Gv(x, z) = Gv(0,z) [1 - Wv(x)] e-(I(z)+n)x, (28)
+ (A - Az + u + (x)) Gv(x,z) = 0, (23)
+ (A - Az + u(x))U(x,z) = 0, (24)
+ (A - Az + 5(x))K(x,z) = 0. (25)
U(x, z) = U(0, z) 1 - Sf (x)
-I(z)-x
(29)
K(x, z) = K(0,z) [1 - Ts(x)] e-I(z) x. (30)
where I(z) = A(1 - z)
Multiplying equation (12) by suitable powers of z, summing over n from 1 to w and using equations (1) and (2) after some algebraic manipulations, we get
fw fw (A + U\
P(0,z) = J Gv(x,z)p.w(x)dx + J Mb(x,z)^n(x)dx -( u J AP00 (31)
Similarly, multiplying equations (14)-(21) by suitable powers of z, summing over n and after some algebraic manipulations, we obtain
a (• w r w 1 /*+w
Mb(0,z) = - K(x, z)S(x)dx + A P(x, z)dx P(x, z) f (x)dx
z 0 0 z 0
r + w + w
+ u / Gv(x,z)dx + U(x,z)u(x)dx, (32)
00
Gv (0, z) = G0,* (0), (33)
(• w
U (0, z) = -/ K(x, z)5(x)dx, (34)
0
K(0, z) = K0(0). (35)
inserting equations 28-29 in 32 we get:
P(0,z) = Gv(0,z)W*(I(z) + n)+ Mb(0,z)G*(I(z)) - (AP00 (36)
in similary way inserting equations 27-31 in 33 we get: 1
Mb (0, z) = ^P(0, z)N* (A) + -T* (I (z))K(0, z) + V (z)Gv (0, z) + P(0, z)(1 -N* (A))
+U(0, z)S* (I(z)) (37)
where V(z) = I(zu+u (1 - W** (I(z) + u)) using equations 16 and 1 :
A2
Gv (0, z) = — P00 (38)
u
using equation 18 :
K(0, z) = AP00 (39)
Hadjadj Houari , Arrar Nawel ,Lahcene Yahiaoui
UNRELIABLE M/G/1 QUEUE WITH GENERAL RETRIAL TIME, V 1 ^902,)
WORKING VACATION AND SETUP TIME_V°lume 2°, March 2025
using equation 31 in 35 :
U(0, z)= -Ts* (I(z))K(0, z) (40)
using equation 40 in 41 :
U(0, z)= -AP00 Ts* (I (z)) (41) using equations 39 40 and 42 in 37 and 38 we get
P(°, z)=DH (42)
(43)
Where Nr(z) = AP00zG*(I(z))Ts*(I(z))[(a - 1)uS*f (I(z)) - -u] + u - AG*(I(z))V(z) - A(W*(I(z)+ u) - 1) Dr(z) = uG*(I(z))[z(1 -N*(A))+ N*(A)] - z
-AP00 i [uz (--S*(I(z)) + - + Sf (I(z))) Ts*(I(z))))
Mb(0,z) = dTzT \ +AzV(z) + z (N*(A) - 1)(u - AW*(I(z) + u) + A) I - (u - AW* (I(z) + u) + A) N* (A)
Substituting equations (39-44) in 27-31 .
4. Steady state results
If the system is in steady state condition p < N* (A), the PGFs are as follows:
(I) the number of customers in the orbit when the server is idle;
P(z) = fw P(x, z)dx = P(0,z)(1 -N*(A)) (44)
0A
(II) the number of customers in the orbit when the server is regularly busy;
Mb(z) = jT Mb(x,z)dx = Mb(0,z)(1i{~ G*(I(z)) (45)
(III) the number of customers in the orbit when the server is at a lower speed service;
Gv(z) = [ G„(x, z)dx = Gvz)%W*J(z) + w) (46)
(IV) the number of customers in the orbit when the server is at repair time;
r ~ U (0, z)(1 -S* ( I (z)) U(z) = Jo U(x, z)dx =-j(z-yl--(47)
(V) the number of customers in the orbit when the server is at setup time;
K(z) = [ K(x,z)dx = K(01 (z)) (48)
From the above equations, the only unknown is P00 which can be obtained by using the normalization condition P00 + Q0 + P(1) + Mb(1) + Gv(1) + U(1) + K(1) = 1 as
P =_w2(A*(A) - AE (G))_
P00 = —,-—1—7-x---rx-(49)
w2 (A [E(Sf )(1 - a) + E(TS)J + N* (A)j j +w (a2 {2W*(w) (AE(Nb) -N* (A) + 1) - E(Nb)W* (w)} +AN*(A) + A2 (1 -W*(w))
Hadjadj Houari, Arrar Nawel ,Lahcene Yahiaoui
UNRELIABLE M/G/1 QUEUE WITH GENERAL RETRIAL TIME, V 1 ^rvM An8,2,) WORKING VACATION AND SETUP TIME_V°lume 20, March 2025
Corollary 1. If the system satisfies the steady state condition, The PGF of the number of customers in the system (Ks(z)) is obtained using
Ks (z) = P00 + Q0 + P(z) + zMb (z) + zGv (z) + U(z) + K(z). (50)
The PGF of the number of customers in the orbit (K0(z)) is obtained using
Ko (z) = P00 + Q0 + P(z)+ Mb (z) + Gv (z) + U (z) + K(z). (51)
5. System performance measures Our analysis is based on the following system characteristics of the retrial queueing system.
5.1. System state probabilities
1. The steady state probability that the server is idle during the retrial time is given by I = P(1)
-AP00 (N*(A) - 1) I w (-E(Nb)A (W* (w) - 1) + w (-*E(Sf) + E(Nb) + E(TS))) = w2(N*(A) - AE (Nb)) j +wE(Sf)+ AwW*(w) + A[wW*'(w) -W*(w) + 1)
2. The steady state probability that the server is busy on normal service period is given by
A ) [ w2 (-aE(Sf)A + E(TS)A + E(Sf)A + N*(A)J
U = Mb(1) = w2(N*(A)-AE(Nb)) j +w (2A2W*(w) - AW*(w)N*(A) + AN*(A)]
I -A2 W* (w) + A2
3. The steady state probability that the server is busy on working vacation period is given by
V = Gv(1) = A2P00 •(1 - W*(w)) w2
4. The steady state probability that the server is under repair time is given by
S = U(1) = E(Sf )AP00 • (1 - a)
5.The steady state probability that the server is under setup time is given by
K = K(1) = E(TS )AP00
5.2. Mean system size and orbit size
(i) The expected number of customers in the orbit (L^ is obtained by differentiating equation 52 with respect to z and evaluating at z = 1
Lq = K0 (1)= ft ^o(z)
(ii) The expected number of customers in the system (Ls) is obtained by differentiating equation 51 with respect to z and evaluating at z = 1
Ls = KS (1)= zhi1 TzKs (z)
(iii) The average time a customer spends in the system (Ws) and the average time a customer spends in the queue (Wq) are found using Little's formula
Ws = A and Wq = .
Hadjadj Houari , Arrar Nawel ,Lahcene Yahiaoui
UNRELIABLE M/G/1 QUEUE WITH GENERAL RETRIAL TIME, V 1 ^902,)
WORKING VACATION AND SETUP TIME_V°lume 2°, March 2025
6. Reliability Measures
In the retrial queueing system with unreliable server, the reliability measures provide the information, which is required for the improvement of the system.
(i) The steady state availability Av, which is the probability that the server is either working for a positive customer or in an idle period such that the steady state availability of the server is given by
Av = 1 - U(1) = 1 - E(Sf )AP00 • (1 - a) (52)
(ii) Let Ff be the steady state probability of server failure,
Ff = aK(1) = aE(Ts )AP00
Theorem 2. Let E(Tb) and E(Tc) be the expected length of busy period and busy cycle under the steady state conditions, we have
w2 (-aE(Sf) + E(Nb) + E(TS) + E(Sf)))
E(Tb) = w2(A* (A) - AE (Nb)) j w (-E(Nb)AW*(w) + 2AW*(w) (E(Nb)A - N*(A) + 1)
j +N* (A) - AW* (w)+ A
(53)
1 [ w2 (A (E(Sf)(1 - a) + E(TS))+N*(A)))
E(Tc) = w2 A(N* W- AE (Nb)) I +w (A^2W* (w) (AE(Nb)-N*(A) + ^J) | (54)
j -A2wE(Nb)W*(w) + AN*(A) + A2 (1 - W*(w)) J
Proof. The result follows directly by applying the argument of an alternating renewal process which leads to
P00 = E (TbE)(+TE(T0); E (Tb) = A (i - 0 and E (Tc) = E (T0) + E (Tb) (55)
where T0 is the time length that the system in empty state. Since the inter-arrival time between two customers follows exponential distribution with parameter A, we have that E (T0) = (1/A). Inserting equation 50 into 56 and by direct calculations, we can get 54 and 55.
7. Cost model
In practice, queue managers aim to minimize the operating cost per unit time. In this section of the paper, we begin by formulating a steady-state expected cost function per unit time, where the service rate ^n is the decision variable. Our objective is to find the optimal value of Цn to minimize the expected cost function. To reach this, We need to define cost elements as follows:
- C1 : is the cost of each consomer in the system per unit of time.
- C2 : represents the cost per unit time to leave the server functioning
- C3 : Cost per service per unit time.
- C4 : represents the cost per unit time needed to prepar starting up the server. Let - Tc be the total expected cost per unit time of the system:
Tc = C1 Ls + C2(1 - P00) + C3Un + C4P00.
7.1. Quadratic Fit Search Method
This part considers the cost optimization problem under a given cost structure via quadratic fit search method (QFSM), This method uses a 3-point pattern to fit a quadratic function that ensures a unique optimal solution., see [22] . So, We aim to optimize the service rate u n in various cases to minimize the expected cost function Tc denoted in this part by H. Assume that all system
parameters have fixed values, and the only controlled parameter is the service rate Fn. Thus, The optimization problem can be mathematically expressed as:
Minimize : H) = CiLs + C2(1 - P00) + C+ C4P00.
As it has been mentioned in Laxmi et al [23], given a 3-point pattern, we may fit a quadratic function via corresponding functional values that has a unique minimum, xq, for the given objective function H(x). the quadratic fit improves the current 3-point pattern by replacing one of its points with the optimum xq , using this approximation. The unique optimum xq of the
quadratic function agreeing with H(x) at 3-point operation (^x1, xm, xu^ is given as
((xm)2 - (xu)2) + H (xm)(^(xu)2 - (x1^ ^ + H (xu) ^ (x1^ 2 - (xm)2^) H (x1 ) (xm - xu) + H (xm) (xu - x1 ) + H (xu) (x1 - xm)
For the whole analysis in this numerical part, we fixe Ci = 10, C2 = 350, C3 = 20, C4 = 120,.
q1
xq C^ _
x = 2
7.2. Optimization analysis
To conduct the numerical analysis for parameter optimization in the queueing system under consideration, we use the following default values for the parameters: a = 0.7, A = 2,5 = 8, fw = 2, s = 5, r = 4 and £ = 8, and the tolerance of QFSM is e = 10-6.
From Figure 1, The curves clearly show convexity, which means , the existence of a specific service rate fn that minimizes the total expected cost function for the given set of model parameters. By adopting QFSM and choosing the initial 3point pattern as , ^, Fn^ = (3.05,3.5,3.75) , and after finite iterations, we observe that the minimum expected operating cost per unit time converges to the solution H = 372.29 at F*n = 3.282,
400 -
395 -390 -
£
385 -380 -375 -
2.5 3.0 3.5 4.0 4.5 5.0
normal rate
Figure 1: The optimum service rate ^
Moreover, we examine the behavior of the expected cost function under different values of the cost parameters. System parameters are fixed as follows: a = 0.7, A = 2,5 = 8, Fv = 2, s = 5, r = 4 and £ = 8; Tables 1-3 illustrate the effects of (C1, C2), (C2, C4) and (C4, C3) on the expected cost function, respectively. It can be see that the expected cost function shows a linearly increasing trend with increasing cost parameters.
Table 1: Effects of (C4, C3) on the expected cost function Tc with Ci = 10 and C2 = 350
(C4, C3) (100,10) (200,10) (200,15) (120,5) (120,20) T 272.4873 313.8563 323.8563 270.7611 300.7611
Table 2: Effects of (C2, C4) on the expected cost function Tc
(C2, C4) (350,150) (350,200) (250,200) (150,120) (100,120) Tc 293.1718 313.8563 255.2253 163.4991 134.1836
Table 3: Effects of (C4, C3) on the expected cost function Tc
(C4, C3) (100,10) (200,10) (200,15) (120,5) (120,20) Tc 272.4873 313.8563 323.8563 270.7611 300.7611
8. Sensitivity Analysis and Numerical Examples
In this section, we provide numerical examples using Python to illustrate how different parameters affect system performance measures. We assume that retrial times, service times, lower-speed service times, vacation times, setup times, and repair times all follow exponential distributions. The parameter values are chosen to satisfy the system's stability conditions. The following tables present computed values for various model characteristics, such as the probability that the server is idle (P00), the mean orbit size (Lq), probability that server in working vacation , probability that server in setup time, and probability that server in repair time . The exponential distribution
is k(x)
ve
x, x > 0.
In Figure 2 , we examine the behavior of the idle probability (P00) increases for increasing the value of the lower service rate (^n) and regular service rate (^w).
In Figure 3 , we examine the behavior of the mean orbit size (Lq) decreases for increasing the value of lower speed service rate ^w and retrial rate
Figure 2: Variation in Pqq with ^n and ^
Figure 3: (Lq) versus £ and
w
In Figure 4 , we see that the behavior of the mean orbit size (Lq) decreases as the values of the lower service rate ^w and regular service rate ^n increase.
In Figure 5 , we examine the behavior of the idle probability (P00) increases with an increase in the setup rate, for a fixed value of repair rate.
Figure 4: (Lq) versus ^n and ^w
Figure 5: P00 versus setup rate and repair rate
Figure 6 depicts that with an increase in repair rate, the probability of the server being in repair state decreases.
Figure 7 depicts that with an increase in service rate Fn , the probability of the server being in setup state increases ;this is due to faster activation of server with reduced setup time.
Figure 6: Effect of repair rate on repair state probability of server
Figure 7: Probability of server in set up versus ^ n
We observe from Figure 8 that the probability of the server being in vacation state decreases with an increase in the rate of working vacation. The reason behind the observation is a decrease in the duration of vacation with an increase in the vacation rate.
Figure 8: Probability of server in vacation versus vacation rate
9. Conclusion
In this article, we analyzed an Unreliable single-server retrial queue model with general retrial time, working vacation and setup time. if certain required and sufficient conditions are satisfied the system can be stabilized .Using he supplementary variable approach and the Probability Generating Function (PGF) approach to determine The PGF of the no. of clients in the system and its orbit. The performance of the model is illustrated using PYTHON . The operating cost of the queuing system is optimized by adjusting the service rate of the server.
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