Performance Analysis of Markov Retrial Queueing Model under Admission Control F-Policy

INTRODUCTION

The formation of queues can be observed everywhere in real-life scenarios, including in front of the post office, schools, train ticket counters, shopping malls. For both customers and system organizers, the creation of long queues and service delays are the main issues. In some real-life situations, as well as in the service sectors (e.g., production and manufacturing, software and communications networks, etc.), clients (e.g., calls, employees, queries, notifications, broken equipment, etc.) arrive in a system for operation, but if the service is inaccessible, they are required to quit the service region and are guided to a simulated location called an orbit. Customers from orbit retry for the operation after a random period. Such reattempts within the service give rise to individual queues known retrial queues.

Significant research has been undertaken in the direction of retrial queues, which can be seen in the survey articles [Artalejo, 1999a and 1999b]. Wuchner et al. [2009] suggested that the homogenous multiserver finite source retrial queue be generalized where all performance metrics are checked using the MOSEL-2 tool. Artalejo and

Lopez-Herrero [2012] studied a finite population retrial queueing model using the block-structured state-dependent event (BSDE) approach to investigate the system state’s limiting distribution and waiting time. Wang and Zhang [2013] analyzed the unobservable and observable retrial queue of the single server by examining the two situations with regard to specific information levels. Jain et al. [2015] set up a doubleorbit, finite capacity retrial queue Markov model with an unreliable server to obtain both transient and steady-state probabilities, as well as different performance indices along with cost function.

Very few researchers have incorporated the working vacation (WV) concept to facilitate the queueing analysis of some realistic queueing circumstances. Due to the real-time applications of queueing framework with a WV from the server point of view, such an analysis has drawn the interest of several researchers in the queues during the last decade. Servi and Finn [2002] first proposed the idea of a WV where the server performs the service at a reduced pace, rather than stopping the service altogether over the vacation time. Wang et al. [2009] evaluated the Markovian machining system with a WV using MAPLE software to calculate the queue size distribution and many other system metrics. Jain and Jain [2010] addressed an unreliable Markovian queue witli vacation and established various system indices using the geometric matrix approach. Yang and Wu [2015] analyzed an unreliable Markovian queueing problem operating under N-policy with WV and established matrix-form expressions for stationary probability distribution for mean queue length and other system metrics. Sethi et al. [2019] conducted the machining system’s transient state analysis incorporating the feature of F-policy and WV. Recently. Jain et al. [2020] investigated the repairable machining system by combining modified server vacation principles along with imperfect recovery. The analytical results are obtained by using the recursive technique after including the supplementary variable. Furthermore, the quasi-Newton approach was used in this research work to determine the optimum parameters by reducing the overall cost.

To ensure the system’s smooth operation and service standard, the customers’ entry in the system should be monitored, and this can be achieved by introducing F-policy admission controls as stated in the queueing literature. Gupta [1995] first proposed the idea of F-policy. As per the F-policy, the arrival of customers into the service sector is halted when the system reached maximum capacity. Furthermore, as the queue size falls to a predefined threshold value “F”, the customers are permitted to join the service sector. Nowadays, it is recognized that the monitoring of arrivals is the most significant issue in real-time scenarios in order to promote the level of service of waiting customers. To control entry of customers, Wang et al. [2007] examined the F-policy finite capacity queueing model. Kumar and Jain [2013] established the distribution of the queue size using a recursive approach for the Markovian queueing model functioning under both F-policy and N-policy. Jain et al. [2017] examined an MRP with a WV and F-policy. They obtained a steady-state queue size distribution by successive overrelaxation (SOR) approach in their analysis. Jain and Meena [2017a] analyzed a fault-tolerant system (FTS) with WV and controllable server and the performance metrics are obtained by using SOR approach in their investigation. Several queue theorists used the Runge-Kutta (R-K) approach to pursue the transient analysis of the queueing model [Jain and Meena, 2017b; Jain and Meena, 2018; Sethi and Bhagat, 2019].

In this chapter, the retrial queueing system is developed with the features of F-policy, WV and balking. The queue size distribution of considered system is computed using the R-K approach. The rest of the chapter is arranged as follows. In section 5.2, we give a description of the model and notations. The model governing equations are stated in section 5.3. In section 5.4, different metrics of the system are described in terms of probabilities of system state. The numerical results and system sensitivity analysis are given in section 5.5. Finally, section 5.6 draws the findings of the model under study by addressing the noble features and the potential direction.

MODEL DESCRIPTION

We formulate the Markovian retrial queueing model in this section for WV operating under admission control policy and start-up period. The F-policy’s purpose is to monitor the entry of customers into the system. The underlying principles and notations used for model creation are described as:

The arriving customer follows a Poisson fashion to enter the system with a parameter X. Due to an overcrowded system, the customers may be discouraged from joining the queue, and it is observed that the customers arriving for the service join the system and balk with probabilities p and p = l-p, respectively. During a normal busy period the customers are served by a single server with mean 1 / p according to an exponential distribution (exp. D). On entry, if the customer finds that the server is busy, he or she is required to wait for the service in the orbit and from orbit customer retries for the service according to an Exp. dist. with parameter y. If the server is not occupied by the customer, the arriving customers get served immediately and leave the system. Once the system reaches its capacity M, the setup time is required to restrict the upcoming customers until the number of customers ceases to the predefined value F. The setup time is considered as follows Exp. dist. with rate e. Furthermore, it is assumed that the server needs a start-up period before enabling customer entry through an Exp. dist. with parameter 0. The server moves to a WV with probability ^ until the device is empty, and it proceeds to do service at a slow pace; with complementary probability 1 - %, the server stays idle. The service times during a WV period follows an Exp. dist. with mean 1 /p„. The first-come-first-served discipline (FCFS) is being pursued to serving the customers.

We construct the model governing equations based on the Markov bi-variate process.

Let us define some notations as:

q(x): Denotes the number of customers in the system at the time x.

£(x): Denotes the status of the server at the time x.

State transition diagram of M/M/l/WV queue with retrial orbit

FIGURE 5.1 State transition diagram of M/M/l/WV queue with retrial orbit.

Now, we describe the server status (see Figure 5.1) as follows:

It is noted that |<;(x), £(x):x>0} is a continuous-time Markov process with state-space

The transient state probabilities of the system states are defined as follows:

<2,.„(x): The probability that at time x there are n (1< n < M) failed machines in the system and the server is in state £(x) = t; 0 < i < 5.

TRANSIENT-STATE GOVERNING EQUATIONS

The Markov retrial queueing model’s governing differential-difference equations framed based on the birth-death process are as follows:

1. The differential-difference equations for the normal retrial state have been framed by using the law of conservation of flows as follows:

2. By equating out-flows from the normal busy state with the sum of in-flows from the other state to the normal busy states, the following equations are constructed:

3. To build the differential-difference equations for the normal F-policy, we apply the laws of flow conservation and obtain:

4. The in-flows and out-flows are equated to frame the transient equations as:

5. For the WV states, the following equations are built by implementing flow conservation laws:

6. By equating the in-flows and out-flows equations for the WV F-policy level, we obtain:

We inflict the initial condition as Q( 1) = 1 and (2(0) = 0 for the determination of transient state probabilities by recognizing that the analytical approach to solving the system equation is quite monotonous. We use a numerical method based on the 4th order R-K to solve the equation system, Eqs. (5.1-5.21).

PERFORMANCE MEASURES

To examine the behavior of the system designed numerous system metrics art defined as follows in terms of the transient probabilities:

1. Expected number of customers in the system at a time x is given as

2. The probability of the server being in F-policy is given as

3. The probability of the server being in normal busy state is given by

4. The probability of the server being in a WV state is given by

5. The probability of the system being in retrial orbit is given as

6. Cost function

The cost elements per unit of the total cost are denoted by:

ch: Holding price of every customer present in the system per unit period cb: Cost per unit time associated to the server is busy in a normal state c„.: Cost per unit time associated to the server is in WV state cf. Cost per unit time associated with service rate during F-policy cv: Cost per unit time associated with service rate during WV c,„: Cost per unit time associated with service rate

By considering the above costs associated with the different system metrics, we are now formulating the cost structure as follows:

NUMERICAL RESULTS

This segment analyzes the impact of various system parameters on specific system metrics such as Ls(T), QB(x), etc. In MATLAB software, we calculate the numerical results by encoding the computer program. The subroutine ode45 is used to solve the system of differential Eqs. (5.1-5.21), of different system states. We set default system parameters for illustration purposes as

Cost set-I: ch = $120, cw = $40, cb = $70, cf = $70, cv = $45, c,„ = $60.

Cost set-II: ch = $120, c„. = $30, cb = $40, cf = $20, c„ = $35, c„, = $30.

The numerical findings are shown in Figures 5.2-5.7 and Tables 5.1-5.4 for conducting the sensitivity study. We have the following results, focused on numerical experiments:

  • 1. Effect of x: From Figures 5.2-5.7 and Tables 5.1-5.4, we see that the mean queue length, probability of busy state and the cost function increase as time goes up. but the probability of server is being in WV state decreases as time increase. It is clear that the impact of time (T) diminished as time goes up, which shows that after a certain time, the system becomes stable as such there is no further change in Ls (x).
  • 2. Effect of X: From Table 5.1, it is observed that as X increases Ls(x) and QB(t) increases but £>,v(x), Qf(x) and <2»(x), decreases. From Figure 5.3, we see that the mean queue length Ls(x) is increasing as X increases; the effect of X is not much remarkable after a certain time period, /. 20.
  • 3. Effect of p: In Table 5.2, as service rate (|i) increases, Ls(x) and Qb(x), probability of customer in retrial orbit £>„(x) decreases with p. Further, the probability that the server is being busy in WV Q„ (x) and F-policy state
L(X) vs. X for varying values of X

FIGURE 5.2 Ls(X) vs. X for varying values of X.

L(т) vs. т for varying values of Ц

FIGURE 5.3 Ls(т) vs. т for varying values of Ц.

LW vs. x for varying values of y

FIGURE 5.4 LSW vs. x for varying values of y.

L(X) vs. X for varying values of p

FIGURE 5.5 LS(X) vs. X for varying values of p.

C(X,g) vs. time X and g

FIGURE 5.6 C(X,g) vs. time X and g.

C(T.g) vs. time т and C(T,|4)

FIGURE 5.7 C(T.g) vs. time т and C(T,|4).

Qt (T) increases with p. From Figure 5.4, it can be seen that the expected number of customers in the system decreases with u which is as per our expectation.

4. Effect of 0: In Table 5.3, we observe that as 0 increases, Ls(T) and QB(X) increases. On the contrary, the probability of a server is in WV Qn.(x), retrial orbit Q„(x) and F-policy state Qf(x) decreases.

TABLE 5.1

Performance Measures for the Varying Value of X

0.3

05

1.891395

0.631113

2.48Е-09

0.368211

6.75Е-04

10

4.27391

0.703563

0.003883

0.249797

0.042757

15

5.001269

0.772667

0.011221

0.182177

0.033934

0.5

5

2.922908

0.702569

4.91Е-08

0.289874

7.56Е-03

10

5.822947

0.810853

0.018364

0.087204

0.083578

15

6.530426

0.903275

0.029593

0.035508

0.031623

0.7

5

3.925209

0.741718

3.27Е-07

0.228371

2.99Е-02

10

6.348627

0.866441

0.030456

0.037471

0.065633

15

6.830805

0.929732

0.042479

0.009517

0.018271

TABLE 5.2

Performance Measures for the Varying Value of p

3

05

1.891395

0.631113

2.48Е-09

0.368211

6.75Е-04

10

4.27391

0.703563

0.003883

0.249797

0.042757

15

5.001269

0.772667

0.011221

0.182177

0.033934

5

5

1.47974

0.500622

2.75Е-08

0.499126

2.52Е-04

10

3.010376

0.555224

0.002141

0.433413

0.009222

15

3.391266

0.591191

0.005429

0.394288

0.009092

7

5

1.208703

0.412773

7.93Е-08

0.587125

1.02Е-04

10

2.260192

0.446997

0.000836

0.549917

0.00225

15

2.420881

0.462023

0.001984

0.533666

0.002327

TABLE 5.3

Performance Measures for the Varying Value of 0

1

05

1.891395

0.631113

2.48Е-09

0.368211

6.75Е-04

10

4.273911

0.703563

0.003883

0.249797

0.042757

15

5.001383

0.772673

0.011237

0.182181

0.033908

2

5

1.891395

0.631114

1.97Е-09

0.368212

6.75Е-04

10

4.298082

0.709813

0.001751

0.250494

0.037942

15

5.042577

0.783836

0.00472

0.181528

0.029916

3

5

1.891396

0.631114

1.59Е-09

0.368212

6.74Е-04

10

4.310819

0.712717

0.000897

0.250874

0.035511

15

5.063381

0.788548

0.002311

0.181208

0.027933

TABLE 5.4

Performance Measures for the Varying Value of у

1

05

1.891395

0.631113

2.48Е-09

0.368211

6.75Е-04

10

4.273911

0.703563

0.003883

0.249797

0.042757

15

5.001383

0.772673

0.011237

0.182181

0.033908

3

5

1.708253

0.707814

2.48Е-09

0.291529

6.57Е-04

10

3.603521

0.783898

0.003389

0.180347

0.032367

15

4.285772

0.819604

0.009094

0.143881

0.027421

5

5

1.616971

0.742618

2.48Е-09

0.256734

6.48Е-04

10

3.352484

0.815069

0.00323

0.152391

0.029309

15

4.013714

0.841728

0.008412

0.12457

0.02529

  • 5. Effect of y: From Table 5.4, we see that as у increases than expected number of customers in the system, the probability of server is being a busy state, shows significant changes as у increases.
  • 6. Effect of p: As the joining probability (p) of the customer’s increases, then increases as time grows. It is noticed from Figure 5.5 that as p increases, the expected number of customers in the system, probability of server is being busy state, remain almost constant, i.e., p does not reveal much impact on various indices.

CONCLUSIONS

We investigated the single-server retrial queueing model by introducing practical aspects such as balking, admission control policy and WV. We have established various performance indices, including mean system size, mean queue length, various system sate probabilities, etc., which are further used to construct the cost function. The applicability of F-policy, along with the WV and balking feature, can be understood in many real-time applications where the optimal F-policy would have improved control of customer admission. The present work may further be extended by considering general service time.

REFERENCES

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Artalejo, J.R. and Lopez-Herrero, M.J. 2012. The single server retrial queue with finite population: a BSDE approach. Operational Research. 12: 109-131.

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Jain, M. and Meena, R.K. 2017a. Markovian analysis of unreliable multi-components redundant fault tolerant system with working vacation and F-policy. Cogent Mathematics. 4: 1-17.

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Wang, J. and Zhang, F. 2013. Strategic joining in M/M/l retrial queues. European Journal of Operational Research. 230: 76-87.

Wang, K.H., Chen, W.L. and Yang, D.Y. 2009. Optimal management of the machine repair problem with working vacation: Newton’s method. Journal of Computational and Applied Mathematics. 233: 449-458.

Wang, K.H.. Kuo, C.C. and Pearn. W.L. 2007. Optimal control of an M/G/l/K queueing system with combined F policy and start-up time. Journal of Optimization Theory and Application. 135: 285-299.

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