Transient Analysis of M/M/1 Retrial Queue with Balking, Imperfect Service and Working Vacation

INTRODUCTION

In routine life as well as commercial/industrial scenarios, it has been observed that the queues of customers/jobs are built up. There are numerous examples of waiting in the queues, in particular when the server is most often busy. Sometimes the arrivals would like to wait in the retrial orbit from where they can make reattempts for getting the service after an arbitrary interval of time. When the server becomes free, then the retrial customer can get the service. If they are not satisfied, then they may demand additional service. The retrial model can be fitted in the queueing situations where customers are allowed to wait in the retrial orbit; such problems are commonly seen at ATMs, hospitals, call centres, fuel stations, restaurants, bank and railway counters, admissions in educational institutes, etc. The retrial queues with a vacationing server have many applications, including in telephone network systems, internet, distributed communication systems, manufacturing systems, etc. [1-3].

The server’s vacation feature has been embedded in many queueing models to investigate the real-time system having vacationing servers. The transient analysis of a Markovian queue with a single server with discouraged customers by conserving the multiple vacation (MV) policy was investigated by Ammar et al. [4]. Servi and Finn [5] first gave the concept of working vacation (WV) to analyze M/M/l model by including the concept that a server can work despite remaining idle during the vacation. Over the years, many researchers have done a lot of work using the WV concept in the queueing model dealing with different congestion problems. Ezeagu et al. [6] proposed the transient analysis to obtain some performance indices of Markovian single-server queue by incorporating the concept of WV and recovery policy. In queueing literature, very few research articles presented the study on the transient behaviour of Markovian queueing models by including the WV concept [7-9].

Balking concept in the queueing model is also an important characteristic from the customers’ view point, and it has an adverse effect on the grade of service of any service system. If the customer notices many other customers already in a queue, then they may have the choice either to enter in the queue or to leave the system. If the customer leaves the system without joining the queue, then it is called balking. Markovian queueing model using the balking behaviour of the customers in transient setup has also been studied in a few research articles [10-12]. The transient analysis with the provision of single vacation in a feedback Markovian queue under interrupted closedown was studied by Azhagappan and Deepa [13]. By including the optimal Д'-policy, Azhagappan and Deepa [14] extended their previous work [13] on the transient Markovian single vacation queue with feedback. Recently, Jain and Rani [15] contributed toward unreliable server retrial queue in Markov setup having the balking and reneging behaviour of the customers.

It is observed in many service systems that the server can work at a slow speed instead of remaining idle, even when availing the WV. The sever switches to working mode when there are no customers, and as such it remains free until some customers join the system during WV. During normal busy mode, the server becomes free as soon as it completes the service and the customers from retrial orbit make reattempts for service. However, the customers may not be satisfied with the slow service rendered by the server during WV, and as such they may demand for additional service. Motivated by this fact, in this chapter we are concerned with the transient study of Markovian queueing model in general setup by including the concepts of WV, imperfect service during WV and balking behaviour of the customers. The remaining findings of the research works are arranged section-wise as follow. In section 2.2, we provide the description of the concerned model by making requisite assumptions. After that in section 2.3, Kolmogorov-Chapman equations for different system states are formulated. In section 2.4, we obtain several queueing indices in terms of transient probabilities. In section 2.5, the cost function is framed. In section 2.6, we discuss the sensitivity analysis after computing the numerical results. Finally, conclusions and future directions of research have been given in section 2.7.

MODEL FORMULATION

In the present study, we consider the WV and imperfect service concepts to develop finite single-server Markovian model for the retrial queue along with the concept of balking behaviour of the customers. The customers arrive in the system in Poisson

State transition diagram

FIGURE 2.1 State transition diagram.

fashion with rate X. On arrival, the customers join the queueing system with probability o. In the regular busy state of the server, the customers are served following exponential distribution (Exp-D) with mean l/p. When the system becomes empty, the WV of the server starts; the duration of WV is governed by Exp-D with mean l/0. In the WV duration, the server can also render service to the customers according to Exp-D with a slower rate y(<|i). During the WV period, the customers are satisfied by the primary service with probability b, and unsatisfied customers opt for the additional service with probability b(= 1 — b). After getting the additional service, which is Exp-D with rate the customer leaves the system forever. During the regular busy period, if the new arrival observes that the server is occupied, then he or she joins the retrial orbit. The customers from the retrial pool can retry following Exp-D with rate nix, where n is the orbit size.

At time t, R(t) and q(t) denote the number of customers present in the queue and status of the server, respectively. The state q(t) = 0 refers that server is in WV mode and the server is free, the state q(t) = 1 refers that server is in WV mode and rendering primary service to the customer, q(t) = 2 refers that the server is providing additional service to the unsatisfied customers during working vacation, q(t) = 3 refers that the server is free while operating in regular busy mode, q(t) = 4 refers that the server is in regular busy mode and rendering service to the customer. Now, {(/?(f), q(f)), f > o} represents a two-dimensional (2D) bivariate stochastic process. Figure 2.1 depicts the incoming and outgoing transition rates of all states of the proposed model.

GOVERNING EQUATIONS

Using the appropriate rates as shown in Figure 2.1, the governing equations for different states are framed as follows:

i. The server being in WV mode but in free state (<;(?) = 0).

ii. The server is in WV mode but in busy state (<;(?) = l).

iii. The server is rendering additional service in WV mode (q(r) = 2).

iv. The server being in normal busy (NB) mode but in free state (q(f) = 3).

v. The server is rendering service during NB mode ((<;(/) = 4)). where, 8„* is the Kronecker delta.

The set of ordinary differential Eqs. (2.1-2.9) are solved by using numerical method viz. IV order Runge-Kutta (R-K) technique. It is noticed that R-K technique can be easily implemented using software like Mathematica. Maple, Matlab, etc. We shall obtain the transient probabilities using routine “ode45” in Matlab. The key performance measures can be derived using the transient probabilities of system states.

PERFORMANCE MEASURES

The key performance metrics namely average queue length, probabilities for different states and throughput at time t are derived in terms of transient probabilities as follows:

i. The mean queue length at time t is

ii. The transient probability for the server being on in WV mode and rendering primary service is

iii. The transient probability for the server being in WV mode and rendering additional service is

iv. The transient probability for the server is in NB mode but in free state

v. The transient probabilities that the server is busy in rendering service in NB mode

vi. Throughput at time t is given by

COST ANALYSIS

The main focus of this study is to frame the cost function TC(t) with consideration of different cost elements and activities. The cost per unit time is defined by

where various cost elements per unit time used are

Cwv : cost incurred when the server is in WV mode and rendering primary service

Cm : cost incurred when the server is in WV mode and rendering additional service

CNF : cost incurred when the server is in NB mode but in free state

CNB : cost incurred when the server is in NB mode but in busy state

Cw : waiting cost of each customers present in the queue

TABLE 2.1

Performance Indices by Varying Parameters о and у

0.5

1

0.6163

0.0352

0.2287

0.2529

46.63

3

0.6558

0.0293

0.1605

0.1726

34.93

5

0.7039

0.0321

0.1230

0.1310

32.61

0.7

1

0.4715

0.0399

0.3208

0.3558

52.98

3

0.5000

0.0307

0.2401

0.2566

36.16

5

0.5551

0.0340

0.1927

0.2027

32.81

0.9

1

0.3595

0.0404

0.3867

0.4427

56.80

3

0.3707

0.0290

0.3034

0.3317

36.42

5

0.4238

0.0323

0.2530

0.2704

32.32

SENSITIVITY ANALYSIS

By taking illustration, we obtain numerical results for the expected queue length, throughput, etc. The computer program is coded using MATLAB software to compute the performance measures and exploring the sensitivity of the system descriptors. Following are the default parameters w'hich we will use for computing the performance metrics:

From Table 2.1, for the fixed value of balking probability a, we notice the decreasing trends in PNF, PNB and increasing trend in Pm with the increments in the primary service rate (y) during WV. For fix value of y, with the increase in balking probability (a), increasing trends in PNF, Pm and decreasing trend in PWv can also observed. From Table 2.2, for a fixed value of о during WV, we notice the increasing trends in

TABLE 2.2

Performance Indices by Varying Parameters a and X

0.5

1.5

0.8127

0.0607

0.0768

0.0497

33.41

1

0.6596

0.0293

0.1929

0.1334

31.62

0.5

0.5483

0.0840

0.2194

0.1493

30.70

0.7

1.5

0.7483

0.0399

0.2922

0.3558

33.96

1

0.5571

0.0841

0.3208

0.3683

31.92

0.5

0.4163

0.1192

0.4467

0.2045

30.88

0.9

1.5

0.6848

0.0553

0.1626

0.0984

33.72

1

0.4679

0.1054

0.3939

0.2876

31.94

0.5

0.3211

0.1420

0.4612

0.3892

30.99

TABLE 2.3

Performance Indices by Varying Parameters о and a

0.5

1

0.7919

0.3069

0.0724

0.1033

36.09

3

0.8085

0.0700

0.0413

0.0958

33.41

5

0.8159

0.0705

0.0279

0.0907

32.57

0.7

1

0.6869

0.0805

0.1021

0.1453

36.44

3

0.7117

0.0824

0.0625

0.1405

33.96

5

0.7241

0.0835

0.0441

0.1360

33.09

0.9

1

0.5824

0.0851

0.1212

0.1769

35.22

3

0.6127

0.0880

0.0789

0.1786

33.72

5

0.7184

0.1004

0.0406

0.1162

33.13

PMr, PNB and decreasing trend in Pm with the increments in the arrival rate (X) during WV. From Table 2.3, we notice that for fix value of u by enhancing balking probability (a), probability that the server is in normal busy period and the server is free (Pnf ), probability that the server is in normal busy mode and busy (PNB) increase but Pwv decreases. From Table 2.4, for a fixed value of o, we see the decreasing trend in Pm, while enhancing the value of probability (b) during WV. In Tables 2.1-2.4, for a fixed o, we have seen that there are minor changes in Pm(t) for varying values of parameters у, X, a, b, respectively.

In Figure 2.2(i-iv), mean queue length £[jV(0] builds up as time t passes; however, initially, it changes slightly, but as time grows, a remarkable increment is noticed. From Figure 2.2(ii and iv), we observe that as arrival rate (X) and probability (b) increase, the queue length £[7V(o] also increases. On the other hand, when we increase the service rates у and p, there is significant decrement in the queue length as clearly noticed in Figure 2.2(i and iii). From Figure 2.2(v), after

TABLE 2.4

Performance Indices by Varying Parameters a and b

0.5

0.4

0.8833

0.0325

0.0605

0.0584

36.09

0.6

0.8537

0.0631

0.0486

0.0495

37.27

0.8

0.8344

0.0922

0.0330

0.0381

38.51

0.7

0.4

0.8705

0.0866

0.0118

0.0250

36.44

0.6

0.8533

0.0737

0.0293

0.0381

38.06

0.8

0.8492

0.0410

0.0562

0.0579

39.81

0.9

0.4

0.7722

0.0474

0.0885

0.0853

35.22

0.6

0.7114

0.1314

0.0604

0.0638

37.00

0.8

0.7051

0.1307

0.0633

0.0661

38.95

Expected queue size E{N(t)} with variation in (i) у (ii) X (iii) g (iv) b (v) a

FIGURE 2.2 Expected queue size E{N(t)} with variation in (i) у (ii) X (iii) g (iv) b (v) a

(vi) e.

time t = 2.5, the queue length (£{N(0}) seems to lower down w'ith the increasing value of retrial rate (a). From Figure 2.2(vi), we observe that until the time t = 2 reaches, the average queue length reveals very less effect for any variation in the value of 0, but later on there is gradually decrement for increasing value of 0.

The significant effects of time and different parameters on throughput are noticed in Figure 2.3(i-iv). From Figure 2.3(i), we observe that beyond time t = 2.4, the throughput decreases significantly by giving increment in y. It is clear from Figure 2.3(ii) that the throughput (TP) quickly increases by the increasing arrival rate (X). Also, for lower value of arrival rate (say X = 0.5), throughput is almost constant. During normal state, as service rate (p) increases, the throughput seems to decrease. The system throughput (TP) also decreases w'ith the increment in the probability b, which is shown in Figure 2.3(iv). Figure 2.4 portrays the cost function that demonstrates the increasing trend as time grows up. As service rate of server in WV mode increases, the cost decreases but effect becomes more prevalent as time passes.

In Figure 2.5, trends of the server being in different states by varying time have been depicted. Based on numerical results, we can infer that by including the service during WV, the cost can be controlled to a prespecified level, and the objective of better grade of service (GoS) can be achieved.

Throughput (TP) with variation in (i) у (ii) A. (iii) p (iv) b

FIGURE 2.3 Throughput (TP) with variation in (i) у (ii) A. (iii) p (iv) b.

Total cost (TC) vs. t

FIGURE 2.4 Total cost (TC) vs. t.

Probabilities for different system states

FIGURE 2.5 Probabilities for different system states.

CONCLUSION

The main focus of this research work is to facilitate the performance indices of the transient model of retrial queue with imperfect service during WV and balking behaviour of the customers. In many applications, including malls, doctor’s clinics, call centres, communication networks, e-ticketing systems, etc., the server can provide service in WV mode at a slower pace, and as such the customers may not be satisfied and may require additional service. The inclusion of realistic features of balking, retrial attempts and option of additional service during the WV period allow our model to deal with real-world congestion problem. The model developed in the general setup can be further modified by including the concepts of unreliable server, optimal control strategy viz. N-policy and/or F-policy, etc.

REFERENCES

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