International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
F-policy Retrial Queue with Unreliable Server and F-policy: A Computational Approach MADHU JAIN* and AMITA BHAGAT** *Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee
[email protected]. in
** Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee
[email protected]
Abstract The present study analyzes the transient analysis of finite population F-policy retrial queue with unreliable server. The customers are called from a system with finite population of ‘M’ customers. The server may break down while providing service to the customers and failed server is sent to the repair facility so as to continue the service of customers. The server takes some time to make preliminary settings before starting the repair process called setup time. The repair is completed following ‘threshold recovery policy’, according to which the repair starts only if a sufficient threshold customers are present in the system. The control policy namely F-policy governs the arrival of the customers to the system. The transient behavior of the system has been analyzed using Runge-Kutta method and various performance measures have been obtained. Moreover, cost function is also formulated using an illustration. Keywords: Retrial queue, Unreliable server, Finite population, F-policy, Threshold recovery, Cost function.
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
1. INTRODUCTION Queues form an obvious part of our real life congestion situations. The queues are visible everywhere from house (within family members for some work) to railway counters, shopping malls etc. Such congestion situations have given a blow to the rise of research in the field of queueing theory from recent past. The areas of telecommunication systems, manufacturing systems, robot manufacturing, and machine repair systems are all governed by the study of queueing theory. The customers wait in queue for their turn to avail the service. But sometimes it happens that a customer being deprived of service rather at first stage due to any of the reason; try to repeat its attempt on a later stage when he finds the server free. Such situations give rise to special queues known as retrial queueing systems. The customer who retry for service is known as retrying/repeated customer and thus retrial queues are formed. A variety of work has been done by a number of researchers in this area due to its significant importance in various real life situations. The detailed account on retrial queues are given in the books on retrial queues by Falin and Templeton [1] and Artalejo & Coral [2]. The relevant literature can be found in the survey articles by Yang and Templeton [3], Artalejo [4-5], Artalejo and Falin [6] and Artalejo [7]. A vast literature on queues with unreliable server is available in the form of research articles and survey papers. Sherman and Kharoufeh [8], Atencia et al. [9] analyzed a system of queues with unreliable retrial queues and obtained various performance indices. Efrosinin & Winkler [10] examined a Markovian retrial queueing system with non-reliable server. An unreliable M/M/2/K queueing system under (N, F) policy with multi optional phase repair has been investigated by Jain et al. [11]. Singh and Jain [12] investigated a single server unreliable queueing system with removable service station. Jain et al. [13] studied vacation queueing model for machining system with two unreliable repairman and obtained various performance indices. The queueing systems are basically modeled to frame new efficient queueing systems which may prove favorable to common man both in terms of time and money. For this purpose, the retrial queueing systems with various control policies originated to control the congestion in the system. In the recent past, control policies like F-policy to control the arrival of the customers in the system, N-policy to control the congestion in service, threshold based recovery to control the rush due to repair have been studied in recent past by a few queue theorists. Wang et al. [14] investigated the optimal management problem of an M/G/1/K queueing system operating under combined F policy and an exponential startup time. Heuristic techniques like Quasi Newton method has been used by Wang & Yang [15] to study the M/M/1/K queueing system with F-policy and unreliable server. Jain & Bhagat [16] investigated a finite population retrial queueing system with threshold recovery and unreliable server with geometric arrivals and impatient customers. The queueing systems with finite population also finds special place in the queueing literature as they corresponds to many realistic situations like machine repair problems. A retrial queue with finite number of sources and identical multiple servers in parallel was studied by Alfa and Isotupa [17]. Krishna Kumar
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
et al. [18] analyzed multiserver feedback retrial queue with finite buffer and constant retrial rate. Jain & Upadhyaya [19] studied a finite buffer multiserver queueing system numerically. Yang et al. [20] explored a finite capacity queue and developed a cost function to search the optimal values of threshold parameters. Recently, Shekhar and Jain [21] investigated finite queueing model with multitasks server and blocking. The present study aims to study a finite population retrial queueing system with unreliable server and Fpolicy to control the congestion due to arrival in the system. Various features namely (i) finite population (ii) threshold recovery (iii) F-policy (iv) retrial queue and (v) unreliable server have been incorporated. Using transient probabilities, a number of performance measures have been derived and cost function is also formulated. The rest of the paper has been organized in the following manner. Section 2 deals with the model description and various notations underlying the model. The practical applications of the model are discussed in section 3. Performance indices are laid in section 4 and cost analysis has been done in section 5. Section 6 deals with the sensitivity analysis and finally conclusions are drawn in section 7. 2. Model Description Retrial queueing models are basically the result of those situations when server is unavailable to serve the customers i.e. the server may either be busy with other customers or might be broken-down due to some reasons. In this investigation, we consider a finite population M/M/1 retrial queue with unreliable server and F-policy. The system has a fix calling population of M customers. The basic assumptions and notations describing the model are as:
2.1 Notations The various notations to be used are as:
N (t)
Number of customers in the system at time t Arrival rate of the customers
γ
Retrial rate of the customers waiting in the orbit
μ1
Service rate when the arrivals are allowed in the system
μ2
Service rate when the arrivals are not allowed in the system
ξ
Startup rate following exponential distribution
α1
Failure rate of the server when maximum population has not been called
α2
Failure rate of the server when maximum population has been called
β1
Repair rate for the failure occurred when the arrivals are not allowed
β2
Repair rate for the failure occurred when the arrivals are allowed
Setup rate following exponential distribution
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
(t )
Status of the server at any time t
1, if the server is busy in rendering service when the arrivals are allowed 2, if the server is busy but the arrivals are not allowed 3, retrial state 4, setup state before repair of the (t) brokendown server when the arrivals are allowed 5, repair state of the server when the arrivals are allowed 6, setup state before repair of the brokendown server when the arrivals are not allowed 7, repair state of the server when the arrivals are not allowed Thus, the state space for the system at any time t can be defined as (t ) (t ), N (t ) .
2.2 Assumptions
The basic assumptions underlying the model are:
The customers arrive in the system in Poisson manner with arrival rate .
If an incoming customer finds the server unavailable for the service, then he waits in the retrial orbit so as to try again for the service with rate γ to avail the service.
The arrival of the customers is managed by using control F-policy according to which after the accumulation of M customers in the system no more customers will be allowed at that moment, the arrival will be allowed only when the number of customers reduces to a sufficient number say F. At this stage, the arrival of the customers will be allowed after a start up time following an exponential distribution with rate ξ.
The service process is completed following exponential distribution on the basis of FCFS with rate μ1 or μ 2 .
The server may fail according to Poisson distribution with rate α1 (α2) while servicing the customers.
In order to maintain the functionality of the server, the broken down server is sent for repair. The server takes some time to make some preliminary settings before starting the repair known as setup time which is exponentially distributed with rate .
The broken down server is sent for repair to become as good as earlier. But when Θ(t) = 2 i.e. arrivals are allowed, the repair starts immediately whereas the repair process follows the concept
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
of threshold recovery for the server failed when
Θ(t)=1.
According to threshold policy, the repair
starts only when a minimum number of customers (threshold value) say, q (≥1) has been accumulated in the system.
The broken down server is repaired with repair rate β1 (or β2) so as to continue the service.
2.3 Governing Equations Now, we frame the mathematical model for the retrial system under consideration on the basis of above defined assumptions and notations. For this purpose, we construct Chapman-Kolmogorov equations for finite population model by using the appropriate transition rates. The equations constructed for the model are as: (i) Level 1: The server being busy in rendering service when arrivals are allowed.
dP1,0 (t ) P2,0 (t ) P3,1 (t ) 1 P1,1 (t ) dt
(
( M 1 ) P1,0 (t )
dP1,n (t ) P2, n (t ) P3, n 1 (t ) 1 P1, n 1 (t ) dt
( M n 1)P1, n 1 (t ) (( M n)
(2)
1 1 ) P1, n (t ), n 1, 2,..., q 1
dP1, n (t ) dt
P2, n (t ) P3, n 1 (t ) 1 P1, n 1 (t ) ( M n 1)P1, n 1 (t ) 1 P5, n (t )
(3)
(( M n) 1 1 ) P1, n (t ), n q, q 1,.., F
dP1, n (t ) dt
P3, n 1 (t ) 1 P1, n 1 (t ) ( M n 1)P1, n 1 (t ) 1 P5, n (t )
(4)
(( M n) 1 1 ) P1, n (t ), n F , F 1,..., M 2
dP1,M 1 (t ) P3, M (t ) 2P1, M 2 (t ) dt 1 P5, M 1 (t )
(5)
( 1 1 ) P1, M 1 (t )
(ii) Level 2: The server being busy but arrivals are not allowed.
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
dP2,0 (t ) 2 P2,1 (t ) 2 P7,0 (t ) dt
(6)
( 2 ) P2,0 (t )
dP2, n (t )
2 P2, n 1 (t ) 2 P7, n (t )
dt
( 2 2 ) P2, n (t ),
(7)
n 1, 2,...., q 1, q, q 1,...., F
dP2, n (t )
2 P2, n 1 (t ) 2 P7, n (t )
dt
( 2 2 ) P2, n (t ),
(8)
n F 1,..., M 1
dP2, M (t ) P1, M 1 (t ) 2 P7, M (t ) dt ( 2 2 ) P2, M (t ),
(9)
( F ( M 1))
(iii) Level 3: Retrial state.
dP3,0 (t )
( M ) P3,0 (t )
dt
dP3, n (t ) ( M n 1)P3, n 1 (t ) dt
(( M n) ) P3, n (t ),
(10)
(11)
n 1, 2,..., ( M 1)
dP3,M (t ) dt
P3, M 1 (t ) P3, M (t )
(12)
(iv)Level 4: Setup state before repair of the broken down state when arrivals were allowed.
dP4,1 (t ) dt
dP4, n (t ) dt
1 P1,1 (t ) ( M b1 ) P4,1 (t )
(13)
1 P1, n (t ) (( M n 1)b1 ) P4, n 1 (t )
(14)
(( M n)b1 ) P4, n (t ), n 2,...., ( q 1)
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
dP4, n (t ) dt
1 P1, n (t )
(15)
(( M n 1)b1 ) P4, n 1 (t ) (( M n)b1 ) P4, n (t ), n q,..., ( M 1)
(v) Level 5: Repair state of the broken server when arrivals were allowed.
dP5,1 (t ) dt
( M )b2 P5,1 (t )
(16)
dP5, n (t ) ( M n 1)b2 P5, n 1 (t ) dt (( M n)b2 ) P5,n (t ), n 2,...., (q 1) dP5, n (t ) dt
(17)
( M n 1)b2 P5,n 1 (t ) P4,n (t ) (( M n)b2 1 ) P5, n (t ), n q, q 1,..., ( M 1)
(18)
(vi) Level 6: Setup state before repair of the broken down state when arrivals were not allowed.
dP6, n (t ) dt
2 P2,n (t ) P6, n (t ),
(19)
n 1, 2,..., M
(vii) Level 7: Repair state of the broken server when arrivals were not allowed.
dP7, n (t ) dt
P6,n (t ) 2 P7,n (t ), n 1, 2,..., M
(20)
3. Practical Applications of the Model The present model under consideration closely relates to various real life situations and hence is applicable to many congestion situations. One of them is discussed below Call centres The working of call centers can also be considered as an analogue to the present model. Call centres are basically set either by private companies or government to deal with the queries or problems related to a particular issue. Since, working of call centres is defined to a particular region or customers hence calls
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
are received from a finite population. A number of calls are received everyday out of which some are received and answered while others remain answered. A call dialed may find the server busy/waiting/ retry later and is hence kept in buffer from where it can try later to be answered. The calls can be transferred from other regions to a particular station due to unavailability of serving agents in that particular area but they can be received/allowed if the number of waiting calls reduces to a sufficient number using F-policy so as to control the burst of arriving calls. The server may also lost connectivity and is found non-working which needs to be repaired so as to continue the service. Restoring connectivity demands both time and money, and therefore such lost systems are usually brought to active state if a sufficient number of calls are in queue to be answered (threshold recovery). The repair process may also need some time to restore settings known as setup time before starting the repair process. Hence, working at call centres can also be visualized as the real life application of our model. 4. Performance Indices To have better accessibility of the model, it is vital to justify the efficiency of a particular mathematical model. The model is ranked well if it outperforms in terms of its indices. Therefore, we list below some of the important performance measures in terms of their transient probabilities as: (i) The probability that the server being busy in providing service to the customers
PB (t )
M 1
M
n 1
n 1
P1,n (t ) P2,n (t )
(21)
(ii) The probability that the system is blocked M
M
M
n 0
n 0
n 0
PL (t ) P2,n (t ) P6,n (t ) P7,n (t ) (22)
(iii) The probability that the broken down server is under setup before repair is
PT (t )
M 1
M
n 0
n 0
P4,n (t ) P6,n (t )
(23)
(iv)The probability that the server starts the repair is M 1
M
n q 1
n q 1
PR (t ) P5, n (t ) P7, n (t )
(24)
(v) Expected queue length at time t, is
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
7 M 1
Ls (t ) nPi , n (t ) i 1 n 1
M
(25)
i 2,3,6,7
Pi , M (t ) (vi)Throughput
at time t is obtained by using
M 1
M
n 1
n 1
TP(t ) 1 P1, n (t ) 2 P2, n (t )
(26)
(vii) Availability of the server at time t, is
Av (t )
M 1
3 M
n 1
i 2 n 1
P1,n (t ) Pi,n (t )
(27)
(viii) Failure frequency M 1
M
n 0
n 0
Ff (t ) 1 P1,n (t ) 2 P2, n (t )
(28)
(ix) Expected waiting time in the system at time t, is
Ws (t ) Ls (t ) / eff (t ) Where,
the
eff (t )
effective
(29) arrival
rate
eff (t )
at
time
t
is
obtained
by
using
M 1
( M n )
n 0
* P1,n (t ) P3,n (t ) b1 P4,n (t ) b2 P5,n (t ) 5. Cost Function
Cost incurred on any system, is the most important issue or factor that addresses the common man in various real life situations. A particular technician or designer always targets to design such systems which are efficient enough in terms of their performance but are also economic in terms of money. The incorporation of new parameters in the queueing systems is to formulate optimal queueing systems in terms of cost which can further be applied over real time congestion situations. In order to qualify a queueing model in terms of its effectiveness, we construct the cost function as:
TC ( F , q) CB PB (t ) Ch Ls (t ) C11 (t ) C2 2 (t ) CS
(30)
CRE CSET CR PR (t ) 749
International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
where, CB
: Cost per unit time when the server is
busy;
Ch : Holding cost per unit time of each customer present in the system; CR C1
: Repair cost incurred per unit time for a broken down server; : Cost for providing service to the customer when the arrivals are allowed;
C2 : Cost for providing service to the customer when the arrivals are not allowed; Cs
: Fixed cost for startup process per unit customer when the customers are allowed to enter;
CRE : Fixed cost incurred for each retrial customer at each time; CSET : Fixed cost for setup process before starting the repair process. Now we evaluate the optimal F* and optimal threshold parameter q* along with minimum respective cost for finite population retrial model using direct search approach. The set of default parameters are taken as:
0.0001, 0.0001, 1 0.04, 2 0.05, 1 2 0.0001, 1 2 0.0008, 0.06, 0.05, b1 b2 0.01. The cost parameters are as:
Ch =100 $, Cb = 200 $, CL =200 $, C1 =50 $, C2 =50 $, Cs = 400 $, CSET = 50 $, CR =200 $, CRE = 200 $ In table 1, for a finite population retrial queueing model with population size M = 8, we compare the total cost for the system by varying F and q.
F /q
1
2
3
4
5
6
2
8261.80
--
--
--
--
--
3
1474.20
1473.90
--
--
--
--
4
366.91
366.61
366.38
--
--
--
5
194.84
194.54
194.31
194.17
--
--
6
171.66
171.36
170.81
170.99
170.92
--
7
171.33
171.04
170.81
170.67
170.60
170.57
Table 1: Total cost for finite population model (M=8)
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
λ
(0.0001)
(0.0005)
(0.0007)
(0.0009)
(0.001)
(0.005)
(0.007)
(0.009)
(0.01)
F
*
4
5
6
6
6
10
10
11
11
q
*
3
4
4
4
5
8
8
10
10
29.99 $
49.40 $
58.37 $
66.88 $
70.96
169.36
$
$
190.25$
203.60$
207.81$
*
*
TC(F ,q )
*
*
Table 2: Effect of λ on optimal parameters (F , q )
It is observed that the minimum cost of $ 170.57 is achieved at extreme point (F*, q*) = (7, 6). *
*
For population size M=12, table 2 displays optimal value of (F , q ) for which the cost is minimum. It depicts the effect of arrival rate ( ) on the optimal parameters F* and q* for different cost sets. As clear from the data displayed in table 5 that the total expected cost TC (F, q) increases as the arrival rate increases from 0.0001 units to 0.01 units. An increase in the number of customers in the system or an increase in the rate of calling population definitely increases the total cost of the system. Moreover, optimal F* increases from 4 to 11 for all the three cost sets. Optimal threshold recovery parameter q* exhibits an increase in its value with an increase in the arrival rate, but its nature seems to be constant with some variations in the arrival rate. The cost analysis for finite population retrial queueing system has also been done by plotting the surface graphs for the total expected cost function TC(t) as shown in figs 1(a-d). In figs 1(a-d) surface graphs are plotted for the trends of total cost against different parameters namely threshold recovery (q), arrival rate ( ), service rate (µ1) and breakdown rate (α1), respectively. As demonstrated in fig. 1(a), the total cost of the system decreases with an increase in q; minimum cost is achieved at the upper limit of range of q. Thus, for the finite population retrial system under consideration, repair process must be started when the maximum number of customers is accumulated in the system. The cost TC(t) increases with an increase in arrival rate
due to the fact that as more customers need
the service and as such the total cost of the system increases. The variation in the cost with service rate (µ1) and breakdown rate (α1) are shown by means of figs 1(c) & 1(d). It is seen that TC(t) decreases from $ 44.79 to $ 27.97 as service rate increases from 0.1 to 0.2, and further increases as service rate increases from 0.2 to 0.3 units. The effect of breakdown rate (α 1) on the total cost is demonstrated by means of fig. 1(d). We notice that an increase in α 1 at a fixed time instant decreases the total cost of the system.
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
1(a)
1(b)
1(c)
1(d)
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
Fig. 1: Effect of (a) q (b) λ (c) µ1 and (d) α1 on the total cost of the finite population system
Overall, we conclude that the retrial queueing model with both finite capacity and finite population is sensitive in terms of the performance measures towards various affecting parameters. The optimized cost function can be utilized in order to determine the optimal threshold parameters and minimum cost to frame optimal and more efficient systems. The state probabilities of the server states, queueing measures including the queue length and reliability measures of a system can also be maintained by controlling various parameters.
6. Sensitivity analysis In the present section, we carry out the sensitivity analysis of various performance indices towards various affecting parameters. The sensitivity analysis of indices seems to be quite effective approach to justify the efficiency of any mathematical system when analytic solution seems to be difficult. For the transient solution of the system of differential equations governing the queueing model, we employ R-K method of fourth order using “ode45” function of MATLAB software so as to determine various performance measures of the system. The set of default parameters are taken as:
0.01, 0.01, 1 0.04, 1 0.05, 1 2 0.01, 1 2 0.001, 0.06, 0.05, b1 b2 0.01, M 8, F 7, q 6
(A) Server state probabilities: The server state probabilities play an important role in the validation of any system. Tables 3 & 4 have been constructed to study the sensitivity of various server state probabilities towards arrival rate
and breakdown rate α. It is quite clear from the table 3 that PB(t), PS(t), PL(t), PT(t) & PR(t) increase
with the growth of both arrival rate
as well as with time t. Table 4
Table 3: Effect of λ on the server state probabilities for the finite population model
λ 0.01
t
PB(t)
PS(t)
PL(t)
PT(t)
PR(t)
Ws(t)
10
0.41949
0.00000
0.00000
0.00193
0.00001
16.83
20
0.53792
0.00000
0.00000
0.00509
0.00012
27.18
30
0.58435
0.00001
0.00003
0.00853
0.00063
35.43
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
0.02
0.03
10
0.62821
0.00000
0.00001
0.00322
0.00023
16.74
20
0.72321
0.00006
0.00028
0.00798
0.00398
28.32
30
0.74934
0.00060
0.00182
0.01315
0.01658
39.26
10
0.73427
0.00001
0.00009
0.00412
0.00177
16.82
20
0.79592
0.00063
0.00260
0.00980
0.02394
30.27
30
0.82137
0.00479
0.01363
0.01605
0.08151
45.54
Table 4: Effect of α on the server state probabilities for the finite population model α 0.003
0.006
0.009
t
PB(t)
PS(t)
PL(t)
PT(t)
Pq(t)
Ws(t)
10
0.41037
0.00000
0.00000
0.00802
0.00001
17.04
20
0.51579
0.00000
0.00000
0.02166
0.00012
27.87
30
0.54922
0.00001
0.00003
0.03701
0.00060
36.88
10
0.39705
0.00000
0.00000
0.01571
0.00000
17.37
20
0.48430
0.00000
0.00000
0.04166
0.00011
29.02
30
0.50045
0.00001
0.00003
0.06999
0.00056
39.27
10
0.38416
0.00000
0.00000
0.02307
0.00000
17.72
20
0.45473
0.00000
0.00000
0.06011
0.00011
30.23
30
0.45600
0.00001
0.00003
0.09932
0.00052
41.86
Table 5: Effect of λ on the availability & failure frequency
Ff(t)
Av(t) λ 0.01
0.02
0.03
t
µ1=0.03
µ1=0.05
µ1=0.09
µ1=0.03
µ1=0.05
µ1=0.09
10
0.91284
0.91149
0.91024
0.00091
0.00091
0.00091
20
0.86411
0.85892
0.85425
0.00090
0.00090
0.00090
30
0.82765
0.81734
0.80816
0.00082
0.00083
0.00081
10
0.88660
0.87425
0.87108
0.00090
0.00087
0.00087
20
0.83601
0.82616
0.81673
0.00085
0.00083
0.00082
30
0.81360
0.79665
0.78024
0.00084
0.00081
0.00078
10
0.86198
0.85720
0.85256
0.00087
0.00086
0.00085
20
0.83621
0.82434
0.81258
0.00084
0.00082
0.00081
30
0.83908
0.81811
0.79758
0.00083
0.00083
0.00079
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
depicts the effect of α on these probabilities. The probability of the server being in busy state P B(t) decreases with an increase of α, which is quite obvious. It is very true that an increase in breakdown rate α makes the server unavailable for the service and hence reduces the probability of the server being in busy state at any time t. Also, P s(t), PL(t) & PR(t) decrease but PT(t) increases with an increase in α. It is noticed that for the higher breakdown rate, there is higher tendency of the server to remain under the set up state; this is due to the fact that the server needs the repair.
(B) Reliability Indices: The effects of breakdown rate α & arrival rate
on Av(t) are displayed in figs 2(a-b). Av(t) decreases as
the breakdown rate α increases from 0.003 to 0.009. It is very realistic that the availability of the server decreases as the failure rate α increases because frequent breakdown of the server makes the system more unavailable for the customers. In fig. 2(b) we notice that Av(t) decreases with an increase in
which
is quite obvious as an increase in arrival rate of the customers makes the system more crowded and less available for the service. The effects of some sensitive parameters on the A v(t) and Ff(t) of the system are displayed in table 5 which reveals the effect of
& µ 1 on the reliability indices with time t. It is clear from
the table that Av(t) decreases with an increase in service rate µ 1 from 0.03 to 0.09 for constant value of t & . We note that the failure frequency decreases but to a very less extent with an increase in the service
Av(t)
rate µ1. However, both Av(t) & Ff(t) decrease as
1.00 0.96 0.92 0.88 0.84 0.80 0.76 0.72 0.68 0.64 0.60 -5
increases.
α=0.003 α=0.007 α=0.009
5
t
15
25
Fig 2(a): Effect of breakdown rate α on Av(t) with time t
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Av(t)
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1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84
λ=0.008 λ=0.009 λ=0.01
0
5
10 t 15
20
25
Fig. 2(b): Effect of arrival rate λ on Av(t) with time t (C) Queueing Indices: It is observed from table 3 that the waiting time W s(t) of the customer in the system increases with an increase in both time as well as the arrival rate ( ). An increase in
directly corresponds to an increase in
waiting time due to an increase in the number of customers in the system. Moreover, waiting time also increases with an increase in t as well as with breakdown rate as displayed in table 4. This observation matches with the fact that if a server is under repair or breakdown state then the customer’s waiting time to get repair increases. With an increase in , the number of customers increases in the system which in turn reduces the throughput. Fig. 3(a) shows that TP(t) decreases as
increases
from 0.007 to 0.01 units. However, opposite trends are visible in Fig. 3(b) for TP(t) with service rate µ 1. More the service rate more is the chance of getting successful services in lesser time and hence more throughput. The sensitivity of queue length Ls(t) towards
& µ1 is depicted in figs 4(a-b). As expected, the
queue length increases (decreases) with an increase in arrival rate (service rate). An increase in service rate from 0.03 units to 0.05 units reduces the number of customers waiting for the service. Therefore, throughput of the system can be increased by increasing the service rate. Also, the number of customers in the system can be controlled by rendering faster service.
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International Conference on Emerging Trends in Global Management Practices – An Interdisciplinary Approach
0.05
λ=0.007 λ=0.009 λ=0.01
TP(t)
0.04
0.03
0.02 0
50
100 t
150
200
3(a) 0.06
µ1=0.03 µ1=0.04 µ1=0.05
0.05
TP(t)
0.04 0.03 0.02 0.01 0.00 0
50
100 t
150
200
3(b) Fig. 3: Effect of λ & µ1 on the Throughput TP(t) of the system 5
µ1=0.03 µ1=0.04 µ1=0.05
Ls(t)
4 3 2 1 0 0
50
100 t
150
200
4(a)
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10
λ=0.010 λ=0.015
Ls(t)
8
λ=0.020
6 4 2 0 0
50
100 t
150
200
4(b) Fig. 4: Effect of µ1 & λ on the queue length Ls(t) of the system
7. CONCLUSION The finite population retrial queueing system with unreliable server has been studied under the threshold based recovery. The customers have the provision to wait in the orbit in order to try again and again for the service at their ease. The enrichment of F-policy makes the system more realistic to real life situations as various control policies can help in reducing congestion in the systems like shopping malls, window tickets etc. Moreover, concept of threshold recovery makes the system more economic and useful as it really helps in minimizing the total cost of system operating under optimal F-policy. In order to determine the optimal parameters, the cost optimization has been done using heuristic approach which may help in system designers and technicians to develop more efficient models based on the information obtained. To visualize the effect of various parameters on the system efficiency and productivity, sensitivity analysis has been performed using numerical illustration. The study of such retrial queueing model based on real life scenarios play a significant role in reducing the congestion by means of optimal control policies, by designing optimal systems with the help of optimal parameters obtained and can also help in keeping track of the various parameters on system’s performance. This study can be further extended by using the arrival of negative customers as well as other control policies like N-policy.
Acknowledgment
The second author is thankful to MHRD for providing financial grant in the form of Senior Research Fellowship (SRF) to carry out the research work. REFERENCES
[1] G. Falin and J. Templeton, Retrial Queues, Chapman & Hall, 1997.
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[2] J. Artalejo and A. Corral, “Retrial Queueing Systems”, Springer, 2008. [3] T. Yang, J. Templeton, “A survey on retrial queues”, Que Syst, vol. 2 no. 3, pp. 201–233, 1987. [4] J. Artalejo, “A classified bibliography of research on retrial queues, Progress in 1990-1999”, Top, vol. 7, no. 2, pp. 187-211, 1999a. [5] J. Artalejo, “Accessible bibliography on retrial queues”, Math Comput Model, vol. 30, no. 3-4, pp. 1-6, 1999b. [6] J. Artalejo and G. Falin, “Standard and retrial queueing systems: A comparative analysis”, Rev Mat Complut, vol.15, no. 1, pp. 101-129, 2002. [7] J. Artalejo, “Accessible bibliography on retrial queues”, Math Comput Model, vol. 51, pp.1071-1081, 2010. [8] N. Sherman and J. Kharoufeh, “An M/M/1 retrial queue with unreliable server”, Oper Res Lett, vol. 34, no. 6, 697-705, 2006. [x]/
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[18] B. Kumar, R. Rukmani and V. Thangaraj, “On multiserver feedback retrial queue with finite buffer”, Appl Math Model, vol. 33, pp. 2062–2083, 2009. [19] M. Jain and S. Upadhyaya, “Threshold N-Policy for degraded machining system with multiple type spares and multiple vacations”, Qual Tech Quant Mgmnt, vol. 6, no. 2, pp.185-203, 2009. [20] D. Yang, Y. Chiang, C. Tsou, “Cost analysis of a finite capacity queue with server breakdowns and threshold-based recovery policy”, J Manuf Syst, vol. 32, no. 1, pp.174-179, 2013.
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