M / M (k) /1 Queuing model with varying bulk service

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International Journal of Mathematics and Soft Computing Vol., No. (0), 09 7. ISSN 49 338 M / M () / Queuing model with varying bul service T.S.R. Murthy Professor, Shri Vishnu Engineering College for Women, Bhimavaram. West Godavari district, Andhra Pradesh, INDIA. E-mail: sriram_pavan3@yahoo.co.in D. Siva Rama Krishna, Assistant Professor, Swarnandhra Institute of Engineering and Technology Narsapuram. West Godavari district, Andhra Pradesh, INDIA. E-mail: srd999@gmail.com G.V.S. Raju Professor, Swarnandhra Institute of Engineering and Technology Narsapuram. West Godavari district, Andhra Pradesh, INDIA. E-mail: drgvsraju@yahoo.com Abstract In this paper, we study single server bul service queuing process in which customers are served in varying batch size and obtain various characteristics of the systems. Keywords: Queuing system, service process, arrival process, bul service, joint probability, marginal probabilities. Introduction The single server models with the interdependent structure had been studied by many authors with single service mechanism. In this paper we consider a generalized queuing model in which customers are served as a batch of size at a time, except when there are less than customers in the system at the time of service.for developing these interdependent models with bul service rule we mae use of the dependence structure given by Rao K.S (986). M / M () / Interdependent queuing model with varying batch size In this section, we consider the single server queuing system having the interdependent arrival and processes with bul service. In this system the interdependence can be induced by considering the dependence structures having a bivariate Poisson distribution of the form P X x, X x / t e ( ) t min.( x, x ) j x j t t t j! (x j)! (x j0 j)! where x,x 0,,,... and 0,μ, min(, ). Here P[X =x, X =x / t] is the joint probability of x arrivals and x services during time t. The marginal distribution of arrival and service are Poisson with parameters and µ respectively. Thus inter arrival time and service time follow negative exponentials distributions of the form e -t and µe -µt respectively where is the mean arrival rate and µ is the mean service rate (Feller 969). Let be the covariance between the number of arrivals and services at time t. This dependence structure turns out to be an independent structure if = 0 (Teicher954). x j () 09

0 T.S.R. Murthy, D. Siva Rama Krishna and G.V.S. Raju 3 Postulates of the Model The postulates of the model with this dependence structure are. The occurrence of the events in non-overlapping time intervals is statistically independent.. The probability that no arrivals and no service completions occur in an infinitesimal interval of time Δt is [( + μ -) t] +O (Δt) 3. The probability that no arrival and one service completion occurs in Δt is (μ ) t + O(Δt) 4. The probability that one arrival and no service completion occurs in Δt is (- ) t + O(Δt) 5. The probability that one arrival and one service completion occurs in Δt is t + O(Δt). This postulate is due to the dependence structure between the arrivals and service completions. 6. The probability that the occurrence of an event other than the above events during Δt is O(Δt) For the given values of and μ the covariance = r between the arrival and the service., where r is the correlation coefficient Let P n (t) be the probability that there are n customers in the system at time t. The difference equations of the model are / P n ( t) = - ( ) P n (t) + ( ) P n ( ) + ( ) (t) / P ( ) = - ( ) P ( ) + ( ) 0 t 0 t In the steady-state the transition equations of the model are n n i ( ) P + ( ) P + ( ) P = 0 ( ) P + 0 ( ) i P i t P n P i ( t) () n = 0 (3) Using the heuristic arguments of Gross and Harries (974), we can obtain the solution of these equations as where r (0,) n P = C r, n 0 and 0 < r < (4) n ( ) D is the root of the characteristic equation - ( ) D + ( ) P n = 0 (5) d and D is the operator. dt 4 Measures and effectiveness The probability that the system is empty is given by P = r (6) 0 For various values of and for given value of and µ we computed P 0 values and are given in Table the values of P 0 for fixed, for various and µ are given in Table 4. The values of r is given for various values of and for fixed values of = and µ = 4 in Table. The values of r for fixed, µ and for varying and are given in Table 3.

M / M () / Queueing model with varying bul service Table : The values of r where = and µ = 4. 0.5 0.487 0.4737 0.4595 0.4444 0.486 0.48 0.3939 0.375 0.3548 0.366 0.3586 0.3507 0.343 0.3333 0.338 0.335 0.305 0.906 0.777 3 0.345 0.336 0.394 0.3 0.345 0.306 0.97 0.875 0.77 0.656 4 0.336 0.330 0.338 0.37 0.3097 0.308 0.93 0.839 0.738 0.68 5 0.3343 0.384 0.3 0.355 0.3083 0.3005 0.9 0.83 0.73 0.6 6 0.3336 0.379 0.37 0.35 0.3079 0.300 0.98 0.87 0.78 0.6 Table : The values of P 0 where = and µ = 4 0.5 0.58 0.563 0.5405 0.5556 0.574 0.588 0.606 0.65 0.645 0.634 0.644 0.6493 0.6577 0.6667 0.676 0.6865 0.6975 0.7094 0.73 3 0.6575 0.6638 0.6706 0.6778 0.6855 0.6939 0.708 0.75 0.73 0.7344 4 0.6638 0.6698 0.676 0.683 0.6903 0.698 0.7068 0.76 0.76 0.737 5 0.6657 0.676 0.6778 0.6845 0.697 0.6995 0.7079 0.77 0.77 0.7379 6 0.6664 0.67 0.6783 0.685 0.69 0.6998 0.708 0.773 0.77 0.738 Table 3: The values of r where = and = 0.. μ 3 4 5 6 7 8 9 0 0.0759 0.585 0.39 0.986 0.360 0.475 0.475 0.57 0.574 0.0693 0.455 0.36 0.758 0.3333 0.387 0.4379 0.486 0.539 0.0637 0.345 0.98 0.563 0.304 0.36 0.409 0.4545 0.4979 3 0.059 0.5 0.847 0.395 0.906 0.3385 0.3839 0.47 0.468 4 0.055 0.68 0.73 0.48 0.73 0.387 0.368 0.409 0.44 5 0.054 0.096 0.67 0.9 0.578 0.30 0.343 0.385 0.49 Table 4: The values of P 0 where = and = 0.. μ 3 4 5 6 7 8 9 0 0.94 0.845 0.768 0.704 0.6399 0.585 0.585 0.4773 0.486 0.9307 0.8545 0.7864 0.74 0.6667 0.68 0.56 0.54 0.468 0.9363 0.8655 0.809 0.7437 0.6896 0.6389 0.59 0.5455 0.50 3 0.94 0.875 0.853 0.7605 0.7094 0.665 0.66 0.573 0.538 4 0.945 0.883 0.87 0.775 0.768 0.683 0.638 0.597 0.5578 5 0.9486 0.8904 0.8373 0.788 0.74 0.6988 0.6577 0.685 0.58 From Tables, 4 and equation (6) we observe that for fixed values of, µ and the value of P 0 increases as increases. As the dependence parameter increases the value of P 0 increases for fixed values of, µ and.

T.S.R. Murthy, D. Siva Rama Krishna and G.V.S. Raju The value of P 0 decreases for fixed values of µ, and as increases. As µ increases, the value of P 0 increases for fixed values of µ, and. If the mean dependence rate is zero then the value of P 0 is same as in the (M / M () / ) model. The average number of customers in the system is The average number of customers in the queue is L q where r is given in equation (4). L s r r r (7) r The values of L s and L q are computed and are given in Tables 5 and 7 for given values of, µ and for various values of and respectively. The values of L s and L q for fixed values of, and for varying values of µ and are also given in Table 6 and 8 respectively. Table 5: The values of L where = and µ = 4. s (8) 3 4 5 6 0.950078 0.900057 0.85039 0.799856 0.750088 0.7000 0.649893 0.6 0.549907 0.57787 0.559089 0.540 0.5045 0.49995 0.47885 0.456664 0.43369 0.40964 0.384466 0.5093 0.506478 0.490 0.47536 0.458789 0.443 0.488 0.403509 0.3836 0.36656 0.506478 0.49983 0.47885 0.4649 0.448646 0.4354 0.4487 0.396453 0.37703 0.356484 0.5078 0.48898 0.47536 0.4609 0.44573 0.49593 0.469 0.3947 0.37556 0.35597 0.5006 0.487874 0.47474 0.459854 0.444878 0.4898 0.403 0.3947 0.37538 0.35504 Table 6: The values of L where = and = 0.. s µ 3 4 5 6 7 8 9 0 0.0834 0.88354 0.3094 0.457 0.56744 0.76738 0.8948.0958.33378 0.07446 0.7075 0.767 0.380834 0.49995 0.63854 0.779043 0.94555.3696 0.068034 0.5540 0.47038 0.34468 0.4506 0.5659 0.69047 0.8338 0.99635 3 0.06699 0.4857 0.654 0.3494 0.40964 0.576 0.633 0.7450 0.880406 4 0.0580 0.346 0.099 0.8999 0.375894 0.46778 0.566907 0.67476 0.79757 5 0.05485 0.309 0.9435 0.68875 0.347346 0.4305 0.5045 0.6685 0.77

M / M () / Queueing model with varying bul service 3 Table 7: The values of L q where = and µ = 4. 0.5 0.46878 0.46357 0.390639 0.355456 0.3488 0.8830 0.55993.5 0.9507 0.87 0.00489 0.894 0.785 0.6665 0.5505 0.4364 0.39 0.904 0.06766 3 0.7843 0.7078 0.680 0.536 0.4489 0.3503 0.568 0.6009 0.066 0.096056 4 0.7078 0.6783 0.5505 0.479 0.38946 0.30454 0.67 0.553 0.033 0.093684 5 0.67878 0.6058 0.536 0.454 0.3743 0.9093 0.059 0.7 0.056 0.093097 6 0.67 0.59974 0.5574 0.44854 0.36978 0.878 0.03 0.47 0.0338 0.09304 Table 8: The values of L q where = and = 0.. μ 3 4 5 6 7 8 9 0 0.00634 0.09854 0.07004 0.7 0.0644 0.9938 0.40648 0.5748 0.76778 0.0056 0.04775 0.05807 0.05034 0.6665 0.44654 0.3443 0.45955 0.604396 0.004334 0.0090 0.048938 0.08838 0.3976 0.0409 0.83047 0.37868 0.493735 3 0.003699 0.07857 0.0484 0.07544 0.904 0.736 0.393 0.380 0.406 4 0.0030 0.05446 0.0369 0.0659 0.0694 0.4908 0.0507 0.786 0.350557 5 0.00785 0.0349 0.0365 0.056975 0.089546 0.985 0.785 0.3535 0.307 From equations (7) and (8) and from the corresponding tables we observe that as increases the values of L s and L q are decreasing and also as increases the value of L s and L q are decreasing for fixed values of other parameters. As the arrival rate increases the values of L s and L q are increasing for fixed values of µ, and. As µ increases the values of L s and L q are decreasing for fixed values of, and. When the dependence parameter = 0 then the mean queue length is same as that of M/M () / model. When = this is same as M/M/ interdependence model. The variability of this model can be obtained as r V (9) ( r) And the coefficient of variation of the model is V C. V (0) where L s and V are as given in equations (7) and (9). L s The value of the variability of the system and coefficient of variation for various values of, for fixed values of, µ are computed which are given Tables 8 and 9. The values of the variability of the system and coefficient of variation for fixed values of, and for various values of, µ are given in Tables 0 and.

4 T.S.R. Murthy, D. Siva Rama Krishna and G.V.S. Raju Table 9: The values of V where = and µ = 4..8576.706.57875.43966.379.9045.0753 0.96 0.85305 0.90547 0.8767 0.8385 0.7938 0.74985 0.7085 0.66506 0.678 0.577448 0.538 3 0.796 0.76998 0.7348 0.7033 0.66977 0.63575 0.60707 0.56638 0.599 0.4945 4 0.76998 0.73605 0.7085 0.679544 0.64998 0.69098 0.586909 0.55368 0.5984 0.483565 5 0.75436 0.78084 0.7033 0.673368 0.644374 0.644 0.5889 0.5504880.4836 0.5658 6 0.750 0.75895 0.699 0.673 0.64794 0.63003 0.588 0.549445 0.55866 0.48048 Table 0: The values of V where = and = 0.. μ 3 4 5 6 7 8 9 0 0.08888 0.383 0.393066 0.606957 0.87945.3045.688075.94403 3.054 0.080004 0.9969 0.345394 0.55869 0.74985.03093.38595.839543.47463 0.0766 0.7955 0.308066 0.463397 0.657 0.88463.70977.5737.974975 3 0.06663 0.6365 0.77864 0.440 0.577448 0.773569.0383.30055.6555 4 0.06588 0.49735 0.595 0.374084 0.579 0.68660 0.8889.30064.4 5 0.057 0.384 0.3073 0.3468 0.467995 0.66807 0.7938 0.99775.457 Table : The values of C.V where = and µ = 4..444.4367.4594.475.500075.57474.5583.593335.63993.678836.65949.66997.6886.7094.7337.757363.785999.888.855035.89763 3.70875.7465.7436.767.78357.807459.8343.8650.90009.940376 4.7465.74049.757363.776.79693.8089.84679.876797.9099.950686 5.79545.7450.767.78039.800998.84.85066.879779.93898.95389 6.73358.74634.76309.7874.8068.8534.857.880776.94599.95366

M / M () / Queueing model with varying bul service 5 Table : The values of C.V where = and = 0.. µ 3 4 5 6 7 8 9 0 3.6977.580.076585.83007.666435.547646.45638.38364.3909 3.798686.663.637.90458.7337.60706.567.434438.375 3.9644.76709.46766.97566.794895.6646.563646.48334.4793 3 4.6935.8847.3684.04337.855035.7878.63953.530333.4645 4 4.6404.9609.40435.093.9397.77369.6656.575438.50380 5 4.408 3.006.4797.7374.9695.80.7094.690.544874 From equations (9) and (0) and from the corresponding tables we observe that as µ increases the variability of the system size decreases and the coefficient of variation increases. For fixed values of µ, and, as increases the variability of the system size increases and the coefficient of variation decreases. We also observe that as increases the variability of the system size decreases and the coefficient of variation increases for fixed values of, µ and. As increases, the variability of the system decreases the coefficient of variation increases. For =0 and =, this model reduces to M/M/ classical model. When =, this model becomes M/M/ interdependent model and for = 0, this model is same as M/M () / model. Average time a customer spends in the system Ws = / (μ- ) () Average time a customer spends in the queue W q = r * W s () Table 3: W s = / (μ- ) where = and µ = 4 then W s = 0.5. Table 4: W s = / (μ- ) where = and = 0.. 3 4 5 6 7 8 9 µ 0 0. 0.5 0.4857 0.66667 0. 0.5 0.333333 0.5 0. 0. 0.5 0.4857 0.66667 0. 0.5 0.333333 0.5 0.090909 0. 0. 0.5 0.4857 0.66667 0. 0.5 0.333333 3 0.083333 0.090909 0. 0. 0.5 0.4857 0.66667 0. 0.5 4 0.07693 0.083333 0.090909 0. 0. 0.5 0.4857 0.66667 0. 5 0.0749 0.07693 0.083333 0.090909 0. 0. 0.5 0.4857 0.66667

6 T.S.R. Murthy, D. Siva Rama Krishna and G.V.S. Raju Table 5: W q = ( r * w s ) where = and µ = 4. 0.5 0.436 0.3685 0.975 0. 0.43 0.059 0.9695 0.875 0.774 0.83 0.793 0.7535 0.75 0.6665 0.69 0.5675 0.55 0.453 0.3885 3 0.75 0.68 0.647 0.6 0.575 0.5305 0.486 0.4375 0.385 0.38 4 0.68 0.65 0.69 0.585 0.5485 0.509 0.466 0.495 0.369 0.34 5 0.675 0.64 0.6 0.5775 0.545 0.505 0.4605 0.45 0.365 0.305 6 0.668 0.6395 0.6085 0.575 0.5395 0.50 0.459 0.435 0.364 0.3 Table 6: W q = r * W s where = and = 0.. µ 3 4 5 6 7 8 9 0 0.008433 0.0983 0.0339 0.049767 0.070 0.04375 0.5767 0.635 0.574 0.006364 0.03379 0.09674 0.05458 0.03086 0.03609 0.04053 0.04686 0.0539 0.005349 0.095 0.06636 0.054 0.06068 0.03037 0.03435 0.03878 0.0484 3 0.004568 0.009678 0.043 0.08543 0.0499 0.0608 0.0973 0.03306 0.0365 4 0.00395 0.008389 0.046 0.0646 0.0963 0.089 0.05987 0.08939 0.0376 5 0.003443 0.00734 0.00899 0.0495 0.0769 0.0077 0.093 0.05556 0.08068 Table 7: P [The system size ] = r - where = and µ = 4. 0.5 0.487 0.4737 0.4595 0.4444 0.486 0.48 0.3939 0.375 0.3548 3 0.33956 0.8594 0.99 0.769 0.089 0.04846 0.0988 0.09506 0.0844 0.077 4 0.04077 0.03800 0.03574 0.033448 0.0307 0.0868 0.065 0.03764 0.0 0.087 5 0.0776 0.0888 0.00993 0.00098 0.009 0.00896 0.00739 0.006496 0.0056 0.0047 6 0.00475 0.0038 0.00347 0.0036 0.00785 0.0045 0.006 0.0085 0.005 0.00 7 0.00378 0.0043 0.0008 0.000977 0.00085 0.00073 0.00067 0.0005 0.0004 0.0003 5 Conclusion In this paper we extend the single server interdependent queuing model to bul service queuing model with varying bul service. These models are having wider applicability in transportation, inventory control, machine interference problems, neurophysiological systems and lie for efficient design and to predict the system performance measures. In this paper, the behavior of the system is

M / M () / Queueing model with varying bul service 7 analyzed using the system characteristics lie average number of customers in the system, average number of customers in the queue, variability of the system size, probability of the system emptiness and coefficient of variation of the system and also the average time of a customer spends in the system and the queue. It is observed that the positive dependence between the arrival and service completions can reduce the mean queue length and variability of system size. Acnowledgment The authors are thanful to the referees for their valuable comments and suggestions which have helped in improving the quality of this paper. References [] Attahiru Sule Alfa, Time Inhomogeneous Bul Server Queue in Discrete Time A Transportation Type Problem, Opern. Res., 30. [] K. L. Aurora, Two Server Bul Service Queuing Process, Opern Res., (964). [3] N.T.J. Baiten, On Queuing Processes with Bul Service, J.Roy.Soc., B-6(954). [4] U.N. Bhat, On Single Server Queuing Process with Binomial Input, Opern. Res.,(964). [5] S.C. Borst et.al., An M/G/ Queue with Customer Collection, Stochastic Models., 9(993). [6] Conolly, The Waiting Time Process for a Certain Correlated Queue, Opern. Res., 6(968). [7] F. Downton, Waiting Time in Bul Service Queues, J.Roy.Stat.Soc., 7(955). [8] W. Feller, An Introduction to Probability theory and Its Applications, Vol.II Wiley, New Yor, [9] J.K. Goyal, Queues with Hyper Poisson Arrivals and Bul Exponential Service, Metria,..(967). [0] Gross, Harris, Fundamentals of Queuing Theory, John Wiley & Sons, New Yor, (974), [] N.K. Jaiswal, Bul Service Queuing Problem, Opern. Res., 8(96). [] B.R.K. Kashyap, The Double Ended Queue with Bul Service and Limited Waiting Space, Opern. Res., 4, (966). [3] J. Medhi. and A. Borthour, On a Two Server Marovian Queue with a General Bul Service Rule, Catiers Centre Etudes Rech. Operat. 4(97). [4] T.S.R. Murthy, Some Waiting Line Models with Bul Service, Ph.D. Thesis Andhra University. Visahapatnam, (993). [5] K.S. Rao, Queues with Input Output Dependence, VIII ISPS Annual Conference, held at Kolhapur, (986).

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