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M/M/1 retrial queueing system with pre-emptive priority service and standby server.

Introduction

Queueing systems in which arriving customers who find all servers and waiting positions (if any) occupied may retry for service after a period of time are called Retrial queues.Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication networks, computer networks and computer systems. The detailed survey of retrial queues and bibliographical information have been obtained from Yang and Templeton (1987), Falin and Templeton (1990), Kulkarni and Liang (1996), monograph by Falin and Templeton (1997) and Artalejo (1999a, 1999b). Because of the complexity of the retrial queueing models, analytic results are generally difficult to obtain. There are a great number of numerical and approximations methods are available, in this paper we will place more emphasis on the solutions by Matrix geometric method. The bibliographical information of matrix geometric methods has been obtained from Gomez-Corral. A (2006). Choi and Park (1990) investigated an [M.sub.1], [M.sub.2]/G/l retrial queue with two types of calls, infinite priority queue (or infinite waiting room) for Type I calls and infinite retrial group for Type II and derived the joint generating function of the number of calls in the two groups and the mean queue lengths by supplementary variable methods. Falin, Artalejo and Martin (1993) extended Choi and Park's results to the case where two types of calls may have different service time distributions. Choi and Chang (1999) have studied the Single Server Retrial Queues with Priority Calls. Ayyappan et al (2009) studied the priority services with breakdown and repair of services under pre-emptive priority services In this research paper the work of Ayyappan et al (2009) is extended for Retrial queueing with standby server under priority services.

Description of the Queueing System

Consider a single server retrial queueing system with pre-emptive priority service and standby server in which two types of customers arrive in a Poisson process with arrival rate [[lambda].sub.1] for low priority customers and [[lambda].sub.2] for high priority customers. These customers are identified as primary calls. In this model there are two types of servers (i.e) Basic Server and Standby server.The service times follow an exponential distribution with parameters [[mu].sub.1] and [[mu].sub.2] for both types of customers(by both servers) respectively. We assume that the breakdown and repair of service may occur only for basic server.The breakdown of service of Basic server follows an exponential distribution with parameter [alpha] and repair of service follows an exponential distribution with parameter [beta]. The retrial is introduced for low priority customers only. Let k be the maximum number of waiting spaces for high priority customers in front of the service station.

Description of Standby Server

The server states of basic server are idle or busy or in breakdown and the server states of standby server are idle or busy. The standby server will function only if the basic server is in breakdown and further it is assume that no breakdown for standby server. Both servers follow the Pre-emptive priority service principle. The services by standby server will be governed by the following principles

1. If the breakdown occurs for basic server during his service to low priority customer, then the remaining service of this low priority customer will be done by standby server and after completion of service this low priority customer leaves the system.

2. If the breakdown occurs for basic server during his service to high priority customer, then the remaining service of this high priority customer will be done by standby server and after completion of service this high priority customer leaves the system.

3. The standby server continues his service according to pre-emptive priority principle till the basic server returns to the system.

4. At any time the basic server returns to the system(after repair of service), in this case the standby server immediately handing over the customer (low or high) who is servered by him during this time to basic server and he becomes idle.

If the basic server is free or the basic server in breakdown and the standby server idle, then at the time of a primary call arrival (low/high), the arriving call begin to be served by standby server immediately and leaves the system after completion of the service. If the basic server is busy with either low or high priority customer then the arriving low priority customer goes to orbit and becomes a source of repeated calls. The pool of sources of repeated calls may be viewed as a sort of queue. Every such source produces a Poisson process of repeated calls with intensity a. If an incoming repeated call finds the basic server free or basic server in breakdown and standby server is free, then it is served immediately either by basic server or standby server respectively and leaves the system after service completion while the source which produced this repeated call disappears.

If there is a breakdown in service for a low priority customer during his service (active breakdown), the basic server goes to the state of breakdown and the remaining service of this low priority customer will be served by standby server. If there is a breakdown for a high priority customer during his service (active breakdown), the basic server goes to the state of breakdown and the remaining service of this high priority customer will be served by the standby server.

If any one of the waiting spaces is occupied by the high priority customers then the low priority customers (as a primary call) can not enter into service station and goes to orbit. If any one of the servers is busy and there are some waiting spaces then the high priority customer can enter into the service station and waits for his service. If there are no waiting spaces then the high priority customers can not enter into the service station and will be lost for the system. Otherwise, the system state does not change.

If the server is engaging with low priority customer and at that time the higher priority customer enters then the high priority customer will get service only after completion of the service of low priority customer who is in service. This type of priority service is called the pre-emptive priority service. This kind of priority service is followed in this paper.

Retrial Policy

Most of the queueing system with repeated attempts assume that each customer in the retrial group seeks service independently of each other after a random time exponentially distributed with rate a so that the probability of repeated attempt during the interval (t, t +[DELTA]t) given that there were n customers in orbit at time t is n[sigma] [DELTA]t + O([DELTA]t). This discipline for access for the server from the retrial group is called classical retrial rate policy.

The input flow of primary calls (low and high), interval between repetitions, service times, breakdown and repair of service are mutually independent.

Matrix Geometric Methods

Let N(t) be the random variable which represents the number of low priority customers in the orbit at time t and P(t) be the random variable which represents the number of high priority customers in the queue (in front of the service station) at time t and [S.sub.1](t) represents the basic server state at time t and [S.sub.2](t) represents the standby server state at time t. The random process is described as < N(t), P(t), [S.sub.1](t), [S.sub.2](t) >.

N(t) takes one of the values 0,1,2,3,4,.... at time t

P(t) takes one of the values 0,1,2,3,... k at time t.

[S.sub.1](t) = 0 if basic server is idle at time t

[S.sub.1](t) = 1 if basic server busy with low priority customer at time t

[S.sub.1](t) = 2 if basic server busy with high priority customer at time t

[S.sub.1](t) = 3 if basic server in breakdown at time t

[S.sub.2](t) = 0 if standby server is idle at time t

[S.sub.2](t) = 1 if standby server is busy with low priority customer at time t

[S.sub.2](t) = 2 if standby server is busy with high priority customer at time t.

The possible state spaces are

{(u,v,w,x)/u = 0,1,2,3 ...; v = 0; w = 0,1,2; x = 0}U

{(u,v,w,x)/u = 0,1,2,3 ...; v = 1,2,3 ...; w = 2; x = 0}U

{(u,v,w,x)/u = 0,1,2,3 ...; v = 0; w = 3; x = 0,1,2}U

{(u,v,w,x)/u = 0,1,2,3 ...; v = 1,2,3 ...; w = 3; x = 2}

The infinitesimal generator matrix Q is given below

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[#.sub.16]: -([[lambda].sub.1] + [[lambda].sub.2] + [beta] + M[sigma]) [#.sub.17]: -([[mu].sub.1] + [beta]) [#.sub.18]: -([[mu].sub.2] + [beta] + [[lambda].sub.2]) [#.sub.19]: -([[mu].sub.2] + [alpha]) [#.sub.20]: - ([[mu].sub.2] + [beta])

[A.sub.00], [A.sub.nn-1], [A.sub.nn], [A.sub.nn+1] are square matrices of order 2k+6 for n = 1,2,3...

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the capacity of the orbit is finite say M, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let x be a steady-state probability vector of Q and partitioned as x = (x(0), x(1), x(2),...) and x satisfies xQ = 0, xe = 1. (1)

where x(i) = ([P.sub.i000], [P.sub.i010], [P.sub.i020], [P.sub.i030], [P.sub.i031], [P.sub.i032], [P.sub.i120], [P.sub.i132], [P.sub.i220], [P.sub.i232],....,[P.sub.ik20], [P.sub.ik32]) i = 0,1,2,3...

Direct truncation method

In this method one can truncate the system of equations in (1) for sufficiently large value of the number of customers in the orbit, say M. That is, the orbit size is restricted to M such that any arriving customer finding the orbit full is considered lost. The value of M can be chosen so that the loss probability is small. Due to the intrinsic nature of the system in (1), the only choice available for studying M is through algorithmic methods. While a number of approaches is available for determining the cut-off point, M, The one that seems to perform well (w.r.t approximating the system performance measures) is to increase M until the largest individual change in the elements of x for successive values is less than [??] a predetermined infinitesimal value.

Stability condition

Theorem: The necessary and sufficient condition for system to be stable is ([[[lambda].sub.1]/[[mu].sub.1]] + [[[lambda].sub.2]/[[mu].sub.2]]) < 1

Proof: The necessary and sufficient condition for retrial queueing system with priority services to be stable is ([[[lambda].sub.1]/[[mu].sub.1]] + [[[lambda].sub.2]/[[mu].sub.2]]) < 1. The same condition holds for this model also since the standby server takes charge immedietly after the breakdown of basic server and handing over to basic server when he returns after repair. So there will be no break in service in customers's point of view. Hence we conclude that the stability condition remains the same.

Analysis of steady state probabilities

We are applying Direct Truncation Method to find Steady state probability vector x. Let M denote the cut-off point or Truncation level. The steady state probability vector [x.sup.(M)] is now partitioned as [x.sup.(M)] = (x(0), x(1), x(2),... x(M)) [x.sup.(M)] satisfies [x.sup.(M)] Q = 0, [x.sup.(M)] e = 1.

Where x(i) = ([P.sub.i000], [P.sub.i010], [P.sub.i020], [P.sub.i030], [P.sub.i031], [P.sub.i032], [P.sub.i120], [P.sub.i132], [P.sub.i220], [P.sub.i232],.... [P.sub.ik20], [P.sub.ik32]), i = 0,1,2,...,M

The above system of equations is solved exploiting the special structure of the coefficient matrix. It is solved by Numerical method such as GAUSS-JORDAN elementary transformation method. Since there is no clear cut choice for M, we may start the iterative process by taking, say M=1 and increase it until the individual elements of x do not change significantly. That is, if [M.sup.*] denotes the truncation point then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where is an infinitesimal quantity.

Special cases

(1) This model becomes Single server retrial queueing system with pre-emptive priority service if [alpha] [right arrow] 0 and [beta] [right arrow] [infinity].

(2) This model becomes Standard single server queueing system with preemptive priority service if [alpha] [right arrow] 0, [beta] [right arrow] [infinity] and [alpha] [right arrow] [infinity].

(3) This model becomes single Server retrial queueing system with breakdown and repair if [[lambda].sub.2] [right arrow] 0 and [[mu].sub.2] [right arrow] [infinity].

(4) This model becomes Single Server Retrial queueing system and results coincide with analytic solutions given by Falin and Templeton for various values of [[lambda].sub.1], [[mu].sub.1], [[lambda].sub.2] [right arrow] 0, [[mu].sub.2] [right arrow] [infinity], [alpha] [right arrow] 0, [beta] [right arrow] [infinity] and k.

Systems performance measures

In this section we will list some important performance measures along with their formula. These measures are used to bring out the qualitative behaviour of the queueing model under study. Numerical study has been dealt in very large scale to study the following measures.

We can find various probabilities for various values of [[lambda].sub.2], [[lambda].sub.2], [[mu].sub.1], [[mu].sub.2], [alpha], [beta], [sigma] and k and the following system measures can be easily study with these probabilities

1. The probability mass function of Basic Server state

Let [S.sub.1](t) be the random variable which represents the basic server state at time t. In this model [S.sub.1](t) takes the values 0, 1, 2, 3

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. The probability mass function of Standby Server state

Let [S.sub.2] (t) be the random variable which represents the standby server state at time t.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. The probability mass function of number of customers(low) in the orbit

Let X(t) be the random variable which represents the number of low priority customers in the orbit.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. The Probability mass function of number of high priority customers in the queue.

Let P(t) be number of high priority customers in the queue at time t. In this model we assume that the capacity of high priority customers in the queue is finite and P(t) takes the values 0,1,2,3....k.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. The Mean number of high priority customers in the queue

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6. The Mean number of low priority customers in the orbit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

7. The probability that the orbiting customer (low) is blocked

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

8. The probability that an arriving customer(low/high) enter into service station immediately

PSI = [[infinity].summation over (i=0)] p(i, 0, 0, 0) + [[infinity].summation over (i=0)] p(i, 0, 3, 0)

Numerical study

MNCO Mean Number of low priority Customers in the Orbit

MPQL Mean Number of high priority customers in front of the service station

[P.sub.10] Probability that the basic server is idle

[P.sub.11] Probability that the basic server is busy with low priority customers

[P.sub.12] Probability that the basic server is busy with high priority customers

[P.sub.13] Probability that the basic server is in breakdown

[P.sub.20] Probability that the standby server is idle

[P.sub.21] Probability that the standby server is busy with low priority customers

[P.sub.22] Probability that the standby server is busy with high priority customers

The values of parameters [[lambda].sub.1], [[lambda].sub.2], [[mu].sub.1], [[mu].sub.2] will be chosen so that it satisfies the stability condition which is discussed in section 6.

Table 1, Table 2, Table 3, and Table 4 show the impact of retrial rate over Mean number of customers in the orbit and Mean number of customers in the high priority queue and we infere the following

* Mean number of customers in the orbit decreases as a increases

* Mean number of customers in the high priority queue increases as k increases

Conclusions

The Numerical study on single server retrial queueing system with standby server under pre-emptive priority service by Matrix Geometric Method have been done in elobarate manner for various values of [[lambda].sub.1], [[lambda].sub.2], [[mu].sub.1], [[mu].sub.2], [sigma], [alpha], [beta] and k

The numerical results were obtained by us coincide with Analytic solutions of single Server Retrial Queueing System with pre-emptive priority service(discussed by Falin and Templeton) for various values of [[lambda].sub.1], [[lambda].sub.2], [[mu].sub.1], [[mu].sub.2], [sigma], ([alpha] [right arrow] 0), ([beta] [right arrow] [infinity]) and k is large

From this numerical study, further we state that when retrial rate is high i.e [sigma] > 8000, these results coincide with standard Single server queueing system with pre- emptive priority service for various values of [[lambda].sub.1], [[lambda].sub.2], [[mu].sub.1], [[mu].sub.2], ([alpha] [right arrow] 0), ([beta] [right arrow] [infinity]) and k is large

The numerical results were obtained by us coincide with Analytic solutions of single Server Retrial Queueing System (discussed by Falin and Templeton) for various values of [[lambda].sub.1], ([[lambda].sub.2] [right arrow] 0), [[mu].sub.1], ([[mu].sub.2] [right arrow] [infinity]), [sigma], ([alpha] [right arrow] 0), ([beta] [right arrow] [infinity]) and k is large

References

[1] Artalejo. J.R (1999a). A classified bibliography of research on retrial queues Progress in 1990-1999 Top 7, 187-211.

[2] Artalejo. J.R(Ed) (1999 b). Accessible bibliography on retrial queues, Mathematical and Computer Modeling 30, pp,,223-233.

[3] Ayyappan.G, Muthu Ganapathi Subramanian.A and Gopal sekar (2009). M/M/1 Retrial Queueing System with Breakdown and repair of service under Pre-emptive priority Service, International journal of computing and applications, Vol 4, No.2, Dec 2009, pp 185-200.

[4] Choi B.D and Y. Chang (1999), Single server retrial queues with priority calls, Mathematical and Computer Modelling, 30, No. 3-4, 7-32.

[5] Falin G.I, Artalejo J.R, Martin.M (1993). On the single server retrial queue with priority customers. Queueing systems 14, 439-455.

[6] Falin, G.I. (1990). A survey of retrial queues. Queueing Systems 7, pp 127- 167.

[7] Falin, G.I. and J.G.C. Templeton (1997). Retrial Queues. Chapman and Hall, London.

[8] Gomez-Corral. A (2006). A Bibliographical guide to the analysis of retrial queues through matrix analytic technique, Annals of Operationas, 141, 163-191.

[9] Kulkarni V.G and Liang H.M (1996). Retrial queues revisited. Frontiers in queueing models, CRC press.

[10] Latouche. G and V. Ramaswamy, (1999), Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM Philadelphia.

[11] M.F. Neuts (1981), Matrix Geometric Solutions in Stochastic Models-An algorithmic Approach, the John Hopkins University Press, Baltomore.

[12] Yang T and Templeton J.G.C (1987). A Survey on retrial queues. Queueing systems, 2, No. 201-233.

A. Muthu Ganapathi Subramanian (1), G. Ayyappan (2) and Gopal Sekar (3)

(1) Associate Professor, Tagore Arts College, Pondicherry, India

(2) Associate Professor, Pondicherry Engineering College, Pondicherry, India

(3) Associate Professor, Tagore Arts College, Pondicherry, India E-mail: csamgs1964@gmail.com, ayyappanpec@hotmail.com, gopsek28@yahoo. co.in
Table 1: System Measures for [[lambda].sub.1] = 10
[[lambda].sub.2] = 5 [[mu].sub.1] = 20 [[mu].sub.2] = 25
[alpha] = 10 [beta] = 100 k = 2 and various values of
[sigma].

Sigma Ocut P10 P11 P12 P13

10 45 0.2942 0.4600 0.1816 0.0642
20 41 0.2946 0.4596 0.1816 0.0641
30 39 0.2950 0.4593 0.1816 0.0641
40 38 0.2953 0.4591 0.1816 0.0641
50 37 0.2955 0.4589 0.1816 0.0640
60 37 0.2957 0.4587 0.1816 0.0640
70 37 0.2958 0.4585 0.1816 0.0640
80 36 0.2960 0.4584 0.1816 0.0640
90 36 0.2961 0.4583 0.1816 0.0640
100 36 0.2962 0.4582 0.1816 0.0640
200 36 0.2969 0.4576 0.1816 0.0639
300 35 0.2972 0.4573 0.1816 0.0639
400 35 0.2974 0.4571 0.1816 0.0639
500 35 0.2976 0.4570 0.1816 0.0639
600 35 0.2976 0.4569 0.1816 0.0639
700 35 0.2977 0.4568 0.1816 0.0638
800 35 0.2978 0.4568 0.1816 0.0638
900 35 0.2978 0.4567 0.1816 0.0638
1000 35 0.2978 0.4567 0.1816 0.0638
2000 35 0.2980 0.4565 0.1816 0.0638
3000 35 0.2981 0.4565 0.1816 0.0638
4000 35 0.2981 0.4565 0.1816 0.0638
5000 35 0.2981 0.4565 0.1816 0.0638
6000 35 0.2981 0.4564 0.1816 0.0638
7000 35 0.2981 0.4564 0.1816 0.0638
8000 35 0.2981 0.4564 0.1816 0.0638
9000 35 0.2981 0.4564 0.1816 0.0638

Sigma P20 P21 P22 MNCO MPQL

10 0.9429 0.0400 0.0171 4.6265 0.0449
20 0.9425 0.0404 0.0171 3.0521 0.0449
30 0.9422 0.0407 0.0171 2.5273 0.0449
40 0.9420 0.0409 0.0171 2.2648 0.0449
50 0.9418 0.0411 0.0171 2.1074 0.0449
60 0.9416 0.0413 0.0171 2.0024 0.0449
70 0.9414 0.0415 0.0171 1.9275 0.0449
80 0.9413 0.0416 0.0171 1.8712 0.0449
90 0.9412 0.0417 0.0171 1.8275 0.0449
100 0.9411 0.0418 0.0171 1.7925 0.0449
200 0.9405 0.0424 0.0171 1.6351 0.0449
300 0.9402 0.0427 0.0171 1.5826 0.0449
400 0.9400 0.0429 0.0171 1.5563 0.0449
500 0.9399 0.0430 0.0171 1.5406 0.0449
600 0.9398 0.0431 0.0171 1.5301 0.0449
700 0.9397 0.0432 0.0171 1.5226 0.0449
800 0.9397 0.0432 0.0171 1.5170 0.0449
900 0.9396 0.0433 0.0171 1.5126 0.0449
1000 0.9396 0.0433 0.0171 1.5091 0.0449
2000 0.9395 0.0435 0.0171 1.4934 0.0449
3000 0.9394 0.0435 0.0171 1.4881 0.0449
4000 0.9394 0.0435 0.0171 1.4855 0.0449
5000 0.9394 0.0435 0.0171 1.4839 0.0449
6000 0.9394 0.0436 0.0171 1.4829 0.0449
7000 0.9393 0.0436 0.0171 1.4821 0.0449
8000 0.9393 0.0436 0.0171 1.4815 0.0449
9000 0.9393 0.0436 0.0171 1.4811 0.0449

Table 2: System Measures for [[lambda].sub.1] = 10
[[lambda].sub.2] = 5 [[mu].sub.1] = 20 [[mu].sub.2] = 25
[alpha] = 10 [beta] = 100 k = 4 and various values of
[sigma].

Sigma Ocut P10 P11 P12 P13

10 46 0.2930 0.4600 0.1827 0.0643
20 41 0.2934 0.4596 0.1827 0.0642
30 39 0.2938 0.4593 0.1827 0.0642
40 38 0.2940 0.4591 0.1827 0.0642
50 38 0.2943 0.4588 0.1827 0.0642
60 37 0.2945 0.4587 0.1827 0.0641
70 37 0.2946 0.4585 0.1827 0.0641
80 37 0.2948 0.4584 0.1827 0.0641
90 37 0.2949 0.4583 0.1827 0.0641
100 37 0.2950 0.4582 0.1827 0.0641
200 36 0.2957 0.4575 0.1827 0.0640
300 36 0.2960 0.4572 0.1827 0.0640
400 36 0.2962 0.4571 0.1827 0.0640
500 36 0.2963 0.4570 0.1827 0.0640
600 36 0.2964 0.4569 0.1827 0.0640
700 36 0.2965 0.4568 0.1827 0.0640
800 36 0.2966 0.4568 0.1827 0.0639
900 36 0.2966 0.4567 0.1827 0.0639
1000 36 0.2966 0.4567 0.1827 0.0639
2000 36 0.2968 0.4565 0.1827 0.0639
3000 36 0.2968 0.4565 0.1827 0.0639
4000 36 0.2969 0.4565 0.1827 0.0639
5000 36 0.2969 0.4564 0.1827 0.0639
6000 36 0.2969 0.4564 0.1827 0.0639
7000 36 0.2969 0.4564 0.1827 0.0639
8000 36 0.2969 0.4564 0.1827 0.0639
9000 36 0.2969 0.4564 0.1827 0.0639

Sigma P20 P21 P22 MNCO MPQL

10 0.9428 0.0400 0.0172 4.6647 0.0497
20 0.9424 0.0404 0.0172 3.0817 0.0497
30 0.9421 0.0407 0.0172 2.5541 0.0497
40 0.9418 0.0409 0.0172 2.2902 0.0497
50 0.9416 0.0412 0.0172 2.1319 0.0497
60 0.9414 0.0413 0.0172 2.0264 0.0497
70 0.9413 0.0415 0.0172 1.9510 0.0497
80 0.9412 0.0416 0.0172 1.8945 0.0497
90 0.9411 0.0417 0.0172 1.8505 0.0497
100 0.9409 0.0418 0.0172 1.8153 0.0497
200 0.9403 0.0425 0.0172 1.6570 0.0497
300 0.9400 0.0428 0.0172 1.6043 0.0497
400 0.9399 0.0429 0.0172 1.5779 0.0497
500 0.9397 0.0430 0.0172 1.5621 0.0497
600 0.9397 0.0431 0.0172 1.5515 0.0497
700 0.9396 0.0432 0.0172 1.5440 0.0497
800 0.9396 0.0432 0.0172 1.5383 0.0497
900 0.9395 0.0433 0.0172 1.5339 0.0497
1000 0.9395 0.0433 0.0172 1.5304 0.0497
2000 0.9393 0.0435 0.0172 1.5146 0.0497
3000 0.9393 0.0435 0.0172 1.5093 0.0497
4000 0.9393 0.0435 0.0172 1.5067 0.0497
5000 0.9392 0.0436 0.0172 1.5051 0.0497
6000 0.9392 0.0436 0.0172 1.5040 0.0497
7000 0.9392 0.0436 0.0172 1.5033 0.0497
8000 0.9392 0.0436 0.0172 1.5027 0.0497
9000 0.9392 0.0436 0.0172 1.5023 0.0497

Table 3: System Measures for [[lambda].sub.1] = 10
[[lambda].sub.2] = 5 [[mu].sub.1] = 20 [[mu].sub.2] = 25
[alpha] = 10 [beta] = 100 k = 6 and various values of
[sigma].

Sigma Ocut P10 P11 P12 P13

10 45 0.2929 0.4600 0.1828 0.0643
20 41 0.2934 0.4596 0.1828 0.0642
30 39 0.2937 0.4593 0.1828 0.0642
40 38 0.2940 0.4590 0.1828 0.0642
50 37 0.2942 0.4588 0.1828 0.0642
60 37 0.2944 0.4587 0.1828 0.0641
70 37 0.2946 0.4585 0.1828 0.0641
80 37 0.2947 0.4584 0.1828 0.0641
90 37 0.2949 0.4583 0.1828 0.0641
100 37 0.2950 0.4582 0.1828 0.0641
200 37 0.2956 0.4575 0.1828 0.0640
300 37 0.2960 0.4572 0.1828 0.0640
400 37 0.2962 0.4571 0.1828 0.0640
500 37 0.2963 0.4570 0.1828 0.0640
600 37 0.2964 0.4569 0.1828 0.0640
700 37 0.2964 0.4568 0.1828 0.0640
800 37 0.2965 0.4568 0.1828 0.0640
900 37 0.2965 0.4567 0.1828 0.0640
1000 37 0.2966 0.4567 0.1828 0.0639
2000 37 0.2967 0.4565 0.1828 0.0639
3000 37 0.2968 0.4565 0.1828 0.0639
4000 37 0.2968 0.4565 0.1828 0.0639
5000 37 0.2968 0.4564 0.1828 0.0639
6000 37 0.2969 0.4564 0.1828 0.0639
7000 37 0.2969 0.4564 0.1828 0.0639
8000 37 0.2969 0.4564 0.1828 0.0639
9000 37 0.2969 0.4564 0.1828 0.0639

Sigma P20 P21 P22 MNCO MPQL

10 0.9428 0.0400 0.0172 4.6666 0.0500
20 0.9424 0.0404 0.0172 3.0832 0.0500
30 0.9421 0.0407 0.0172 2.5555 0.0500
40 0.9418 0.0409 0.0172 2.2916 0.0500
50 0.9416 0.0412 0.0172 2.1332 0.0500
60 0.9414 0.0413 0.0172 2.0277 0.0500
70 0.9413 0.0415 0.0172 1.9523 0.0500
80 0.9412 0.0416 0.0172 1.8957 0.0500
90 0.9410 0.0417 0.0172 1.8518 0.0500
100 0.9409 0.0418 0.0172 1.8166 0.0500
200 0.9403 0.0425 0.0172 1.6583 0.0500
300 0.9400 0.0428 0.0172 1.6055 0.0500
400 0.9399 0.0429 0.0172 1.5791 0.0500
500 0.9397 0.0430 0.0172 1.5633 0.0500
600 0.9397 0.0431 0.0172 1.5527 0.0500
700 0.9396 0.0432 0.0172 1.5452 0.0500
800 0.9395 0.0432 0.0172 1.5395 0.0500
900 0.9395 0.0433 0.0172 1.5351 0.0500
1000 0.9395 0.0433 0.0172 1.5316 0.0500
2000 0.9393 0.0435 0.0172 1.5158 0.0500
3000 0.9393 0.0435 0.0172 1.5105 0.0500
4000 0.9393 0.0435 0.0172 1.5078 0.0500
5000 0.9392 0.0436 0.0172 1.5063 0.0500
6000 0.9392 0.0436 0.0172 1.5052 0.0500
7000 0.9392 0.0436 0.0172 1.5045 0.0500
8000 0.9392 0.0436 0.0172 1.5039 0.0500
9000 0.9392 0.0436 0.0172 1.5034 0.0500

Table 4: System Measures for [[lambda].sub.1] = 10
[[lambda].sub.2] = 5 [[mu].sub.1] = 20 [[mu].sub.2] = 25
[alpha] = 10 [beta] = 100 k = 8 and various values of
[sigma].

Sigma Ocut P10 P11 P12 P13

10 45 0.2929 0.4600 0.1828 0.0643
20 40 0.2934 0.4596 0.1828 0.0642
30 39 0.2937 0.4593 0.1828 0.0642
40 38 0.2940 0.4590 0.1828 0.0642
50 38 0.2942 0.4588 0.1828 0.0642
60 38 0.2944 0.4587 0.1828 0.0641
70 38 0.2946 0.4585 0.1828 0.0641
80 38 0.2947 0.4584 0.1828 0.0641
90 38 0.2948 0.4583 0.1828 0.0641
100 38 0.2950 0.4582 0.1828 0.0641
200 38 0.2956 0.4575 0.1828 0.0640
300 38 0.2960 0.4572 0.1828 0.0640
400 38 0.2962 0.4571 0.1828 0.0640
500 38 0.2963 0.4570 0.1828 0.0640
600 38 0.2964 0.4569 0.1828 0.0640
700 38 0.2964 0.4568 0.1828 0.0640
800 38 0.2965 0.4568 0.1828 0.0640
900 38 0.2965 0.4567 0.1828 0.0640
1000 38 0.2966 0.4567 0.1828 0.0639
2000 38 0.2967 0.4565 0.1828 0.0639
3000 38 0.2968 0.4565 0.1828 0.0639
4000 38 0.2968 0.4565 0.1828 0.0639
5000 38 0.2968 0.4564 0.1828 0.0639
6000 38 0.2969 0.4564 0.1828 0.0639
7000 38 0.2969 0.4564 0.1828 0.0639
8000 38 0.2969 0.4564 0.1828 0.0639
9000 38 0.2969 0.4564 0.1828 0.0639

Sigma P20 P21 P22 MNCO MPQL

10 0.9428 0.0400 0.0172 4.6666 0.0500
20 0.9424 0.0404 0.0172 3.0833 0.0500
30 0.9421 0.0407 0.0172 2.5555 0.0500
40 0.9418 0.0409 0.0172 2.2916 0.0500
50 0.9416 0.0412 0.0172 2.1333 0.0500
60 0.9414 0.0413 0.0172 2.0278 0.0500
70 0.9413 0.0415 0.0172 1.9524 0.0500
80 0.9412 0.0416 0.0172 1.8958 0.0500
90 0.9410 0.0417 0.0172 1.8518 0.0500
100 0.9409 0.0418 0.0172 1.8167 0.0500
200 0.9403 0.0425 0.0172 1.6583 0.0500
300 0.9400 0.0428 0.0172 1.6055 0.0500
400 0.9399 0.0429 0.0172 1.5792 0.0500
500 0.9397 0.0430 0.0172 1.5633 0.0500
600 0.9397 0.0431 0.0172 1.5528 0.0500
700 0.9396 0.0432 0.0172 1.5452 0.0500
800 0.9395 0.0432 0.0172 1.5396 0.0500
900 0.9395 0.0433 0.0172 1.5352 0.0500
1000 0.9395 0.0433 0.0172 1.5317 0.0500
2000 0.9393 0.0435 0.0172 1.5158 0.0500
3000 0.9393 0.0435 0.0172 1.5105 0.0500
4000 0.9393 0.0435 0.0172 1.5079 0.0500
5000 0.9392 0.0436 0.0172 1.5063 0.0500
6000 0.9392 0.0436 0.0172 1.5053 0.0500
7000 0.9392 0.0436 0.0172 1.5045 0.0500
8000 0.9392 0.0436 0.0172 1.5039 0.0500
9000 0.9392 0.0436 0.0172 1.5035 0.0500
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Author:Subramanian, A. Muthu Ganapathi; Ayyappan, G.; Sekar, Gopal
Publication:International Journal of Computational and Applied Mathematics
Date:Jul 1, 2010
Words:5748
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