ANALYZING ROUGH QUEUEING MODELS WITH PRIORITY DISCIPLINE

Transcription

1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu ANALYZING ROUGH QUEUEING MODELS WITH PRIORITY DISCIPLINE M.Thangaraj 1 and P.Rajendran 2 1,2 Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India. 1 mthangaraj51@gmail.com 2 prajendran@vit.ac.in Abstract In this paper, a single server rough queue with infinite capacity priority model is discussed. A new method namely, segment method is proposed for finding the total average rough interval cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline in the priority queuing system using four crisp priority queue models which are constructed from the queuing system. Numerical example is presented to illustrate the proposed method. The information obtained by the segment method about the priority discipline are helped decision makers to take an appropriate decision when they are analyzing rough priority queuing systems. AMS Subject Classification: 60K25, 03E72. Keywords and Phrases: Queuing system, Priority discipline, Total average cost of system inactivity, Rough set theory, Segment method. ijpam.eu

2 1 Introduction Queueing theory was introduced by Erlang [2]. Steady state distribution of single server priority queue was developed by Miller [8]. Drekic and Woolford [1] analyzed a two-class, single-server preemptive priority queueing model with low priority balking customers. Haghighi and Mishev [5] discussed a general parallel finite buffer multi server priority queueing system with balking and reneging. Pardo and David de la Fuente [12] developed two fuzzy queueing models (M/M/1 and M/F/1) with priority-discipline. Ritha and Lilly Robert [16] described a fuzzy priority queueing model using DSW algorithm and obtained the total cost of the given problem. The basic concept of rough set theory and its applications are studied by Pawlak [13], [14]. Palpandi and Geetharamani [9] applied fuzzy set theory to three priority queue using robust ranking technique. Pandian et al. [10] proposed slice-sum method for solving fully rough integer interval transportation problem. Many authors [3, 4, 7, 11, 12, 15, 16] studied different types of priority queueing problems and rough interval problems. Recently, Haviv [6] proposed M/G/1 queue model in which all customers are identical ex-ante but prior to joining the queue, they draw a random (preemptive) priority level. Shanmugasundaram and Venkatesh [17] studied the retrial priority queueing model in fuzzy environment. 2 Crisp Priority Queue Model Consider a single server priority queuing model with infinite population, in which arrival rate λ = λ 1 +λ 2 and service rate µ, where λ i denotes the average arrival rate for units of class priority i, i = 1, 2. Let be the average time in the system for units belonging to class i, i = 1, 2. Let W i denote the average time in the priority discipline queuing system, we must compare the total average cost of system inactivity for the system and C i be the unit cost of inactivity for units in class i, i = 1, 2. In the three types, no priority, preemptive priority and non preemptive priority. Now, the total average cost of system inactivity for no priority (P 0 ), preemptive priority (P 1 ) and non preemptive priority (P 2 ) ijpam.eu

3 are computed as follows: (i) When the model has no priorities, the total average cost of system inactivity, P 0 is given below: P 0 = (C 1 λ 1 + C 2 λ 2 )W, (1) where W = 1 µ λ (ii) When the model has preemptive priorities, the total average cost of system inactivity, P 1 is given as P 1 = (C 1 λ 1 W 1 + C 2 λ 2 W 2 ), (2) where W 1 = λ µ 2 (1 σ 1 )(1 σ 2 ) + λ µ, W 2 = λ µ 2 (1 σ 2 ) + λ µ, σ 1 = λ µ and σ 2 = λ 2 µ. (iii) When the model has non preemptive priorities, the total average cost of system inactivity, P 2 is given as P 2 = (C 1 λ 1 W 1 + C 2 λ 2 W 2 ), (3) where W 1 = 1 µ(1 σ 1 )(1 σ 2 ), W 2 = 1 µ(1 σ 2 ), σ 1 = λ µ and σ 2 = λ 2 µ. 3 Rough Priority Queuing Model Consider a single-server priority queuing model in rough environment with infinite population in which arrival rate and service rate are rough interval numbers. The following assumptions are coupled with the considering model: (i) The number of service channels is one. (ii) The arrival source population and the queue are infinite. (iii) Poisson arrival rate [λ] = [[λ 2, λ 3 ], [λ 1, λ 4 ]] and exponential service rate [µ] = [µ 2, µ 3 ], [µ 1, µ 4 ]] are rough interval numbers with λ 4 < sµ 1. Now, the rough interval traffic intensity [ρ] is given as follows: [ρ] = [λ] = [ [λ 2,λ 3 ], [λ 1,λ 4 ] = [[ρ [µ] [µ 2,µ 3 ] [µ 1,µ 4 ] 2, ρ 3 ], [ρ 1, ρ 4 ]] where ρ 1 = λ 1 µ 4, ρ 2 = λ 2 µ 3, ρ 3 = λ 3 µ 2, and ρ 4 = λ 4 µ 1. Note that ρ i < 1, i = 1, 2, 3, 4. Remark1:If ρ i 1, i = 1, 2, 3, 4, then the queue length of the rough interval queueing model will explode.now, based on the rough interval traffic intensity [ρ], a rough interval priority queue model with rough interval mean arrival rate [λ] = [[λ 2, λ 3 ], [λ 1, λ 4 ]] and rough interval mean service rate [µ] = [µ 2, µ 3 ], [µ 1, µ 4 ]] is separated into four independent crisp single-server priority queues Q 1, Q 2, Q 3 ijpam.eu

4 and Q 4 called lower upper level queue, lower lower level queue, upper lower level queue and upper upper level queue respectively as follows: Q 1 : A single server priority queuing system with the mean arrival rate is λ 1 and the mean service rate is µ 4 ; Q 2 : A single server priority queuing system with the mean arrival rate is λ 2 and the mean service rate is µ 3 ; Q 3 : A single server priority queuing system with the mean arrival rate is λ 3 and the mean service rate is µ 2 and Q 4 : A single server priority queuing system with the mean arrival rate is λ 4 and the mean service rate is µ 1. Now, using (1),(2) and (3), the total average cost of system inactivity for no priority, the total average cost of system inactivity for preemptive priority and the total average cost of system inactivity for non preemptive priority discipline in the system Q i, i = 1, 2, 3, 4 are computed. They are denoted as P 0 (Q i ), P 1 (Q i ) and P 2 (Q i ), i = 1, 2, 3, 4 respectively. Now, since the rough interval priority queuing system is separated into four crisp priority queuing systems Q 1, Q 2, Q 3 and Q 4 the total average cost of system inactivity (P 0 (Q)), the total average cost of system inactivity for preemptive priority (P 1 (Q)) and the total average cost of system inactivity for non preemptive priority discipline (P 2 (Q)) in the system Q are given as follows: P 0 (Q) = [[P 0 (Q 2 ), P 0 (Q 3 )], [P 0 (Q 1 ), P 0 (Q 4 )]] (1) P 1 (Q) = [[P 1 (Q 2 ), P 1 (Q 3 )], [P 1 (Q 1 ), P 1 (Q 4 )]] and (2) P 2 (Q) = [[P 2 (Q 2 ), P 2 (Q 3 )], [P 2 (Q 1 ), P 2 (Q 4 )]] (3) 4 The Segment Method We, now propose a new method namely, segment method for finding the total average cost of system inactivity for no priority, preemptive priority and non-preemptive priority in a single server priority queuing system in rough environment. The proposed method is as follows: ijpam.eu

5 Step 1: Construct four independent crisp single server priority queues Q 1, Q 2, Q 3 and Q 4 from the given rough interval queuing system, Q. Step 2: Using (1), (2) and (3), compute the total average cost of system inactivity in Q 1, Q 2, Q 3 and Q 4 for three types of priority discipline, that is, no priority, preemptive priority and non preemptive priority discipline. Step 3: Compute the total average rough interval cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline in the given queuing system, P 0 (Q), P 1 (Q) and P 2 (Q) respectively using (4),(5) and (6). Now, the solution procedure for finding the total average rough interval cost of system inactivity is illustrated with the help of following numerical example. Example 1: Consider a queuing model with two unit classes of arrival. 15 percentage of arrivals belong to one of the classes ( which will be denoted by A), and remaining 85 percentage are in the other class ( which will be denoted by B). The Poisson arrival rate is the rough interval number [λ] = [[28, 30], [26, 32]]. The exponential service rate for single server is the rough interval number [µ] = [[40, 42], [38, 45]]. The unit cost of inactivity for A class is C A = [[18, 20], [15, 22]] and the unit cost of the inactivity for the class B is C B = [[3, 4], [2.5, 5]]. Determine the total average rough interval cost of system inactivity for the queuing system when there is no priority discipline, preemptive priority discipline and non preemptive priority discipline. Now, using the Step 1. and the Step 2., the total average cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline computed as follows. Now, Q 1 : A queue model with the mean arrival rate λ 1 = 26 and the mean service rate µ 4 = 45. Then, the total average cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline in Q 1 are obtained as follows. P 0 (Q 1 ) = 5.98, P 1 (Q 1 ) = 7.42 and P 2 (Q 1 ) = 8.46 Now, Q 2 : A queue model with the mean arrival rate λ 2 = 28 and the mean service rate µ 3 = 42. Then, the total average cost of system inactivity for no priority, preemptive priority and non pre- ijpam.eu

6 emptive priority discipline in Q 2 are obtained as follows. P 0 (Q 2 ) = 10.5, P 1 (Q 2 ) = and P 2 (Q 2 ) = Now, Q 3 : A queue model with the mean arrival rate λ 3 = 30 and the mean service rate µ 2 = 40. Then, the total average cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline in Q 3 are obtained as follows. P 0 (Q 3 ) = 19.2, P 1 (Q 3 ) = and P 2 (Q 3 ) = Now, Q 4 : A queue model with the mean arrival rate λ 4 = 32 and the mean service rate µ 1 = 38. Then, the total average cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline in Q 4 are obtained as follows. P 0 (Q 4 ) = 40.26, P 1 (Q 4 ) = and P 2 (Q 4 ) = Now, using the Step 3., we obtain the following results: (i) The total average rough interval cost of system inactivity for no priority discipline[p 0 (Q)] in the given queuing system is given below: [P 0 (Q)] = [[P 0 (Q 2 ), P 0 (Q 3 )], [P 0 (Q 1 ), P 0 (Q 4 )]] = [[10.5, 19.2], [5.98, 40.26]] (ii)the total average rough interval cost of system inactivity for preemptive priority discipline [P 1 (Q)], in the given queuing system is given below: [P 1 (Q)] = [[P 1 (Q 2 ), P 1 (Q 3 )], [P 1 (Q 1 ), P 1 (Q 4 )]] = [[14.42, ], [7.42, 69.11]] and (iii)the total average rough interval cost of system inactivity for non preemptive priority discipline[p 3 (Q)], in the given queuing system is given below: [P 2 (Q)] = [[P 2 (Q 2 ), P 2 (Q 3 )], [P 2 (Q 1 ), P 2 (Q 4 )]] = [[16.38, 31.86], [8.46, 74.52]] Now, comparing the above three results, we observe that the minimum of the total average rough interval cost of system inactivity for no priority discipline is achieved. Therefore, we can conclude that the given queuing system with no priority discipline is fine since it provides less the total average cost of system inactivity than all other priority disciplines. 5 Conclusion A single server queue with infinite capacity priority model in rough environment is considered in this paper. The considering rough ijpam.eu

7 queue model is segmented into four crisp priority queue models. A new method namely, segment method has been proposed for finding the total average cost of system inactivity for no priority, preemptive priority and non preemptive priority discipline in the considering priority queuing system using its four crisp priority queue models. The proposed method is illustrated using a numerical example. The proposed method provides most important information about the priority discipline to decision makers when they are analyzing queuing systems with priority discipline in rough environment. Acknowledgement: The authors thank Dr.P.Pandian, Professor, Department of Mathematics (SAS), VIT University, Vellore for his encouragement and support in fabricating this paper. References [1] S.Drekic and D.G. Woolford, A preemptive priority queue with balking, European Journal of Operational Research, 164 (2005), [2] A.K.Erlang, The theory of probabilities and telephone conservations, Nyt Jindsskri math., 20(1909), [3] R.Groenevelt and E. Altman, Analysis of alternating-priority queueing models with (cross) correlated switchover times, Queueing Systems, 51(2005), [4] D.Gross, J.F.Shortle, J.M.Thompson and C.M. Harris, Fundamentals of queueing theory, 4th Edition, Wiley, New York( 2008). [5] A.M.Haghighi and D.P. Mishev, A parallel priority queueing system with finite buffers, Journal of Parallel and Distributed Computing, 66(2006), [6] M.Haviv, The performance of a single-server queue with preemptive random priorities, Performance Evaluation, 103(2016), Applied Mathematical Modeling, 35 (2011), ijpam.eu

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