ANALYSIS OF TRI-CUM BISERIAL BULK QUEUE MODEL CONNECTED WITH A COMMON SERVER

The present paper demonstrates the tri-cum biserial bulk queue model linked with a common server with fixed batch size. The development of the model has been done in the steady-state condition. The arrival and servicing patterns of the customers are postulated to follow the Poisson law. Various queuing model performances have been assessed by using the probability generating function technique and other statistical tools. The broad parametric examination has been documented to show the adequacy of the current arrangement procedure.


INTRODUCTION
Queuing theory is an assortment of mathematical models of several queuing systems. The formation of a queue is a natural phenomenon. We face this problem in our daily routine life 2600 SACHIN KUMAR AGRAWAL, VIPIN KUMAR, BRAHMA NAND AGRAWAL everywhere. Several queuing models have been established so far, which enable the individual to take the precise choice in actual time circumstances. These models are viable while managing the practical issues underway businesses, banking parts, and business shopping centers, etc. Many investigations have been accomplished in the past, which dealt with the characteristics of queuing models.
A.K. Erlang developed the concept of queuing theory. Erlang [1] executed this hypothesis to analyze the impact of the fluctuating help request on the use of phones during the discussion.
Suzuki [2] explored the queuing framework comprise of two queues in the arrangement. In the investigation, commonly autonomous irregular factors with the particular circulation work have been utilized to exhibit the administration time at all the administration counters. Maggu [3] explored the numerous waiting line parameters of the waiting line model with phase-type service.
Sharma and Sharma [4] accomplished a specific capacity queuing model with a time-dependent analysis. They have expected that the bulk appearance rate relies upon the kind of administration accessible in the framework. Krishnamoorthy and Ushakumari [5] calculated several queue characteristics of the Markovian queuing model using Little's method in the formation of various governing equations.
Singh et al. [6] studied the transient behavior of a queuing model in which servers were arranged in parallel in a bi-series way. Kumar et al. [7] investigated various queuing parameters of a complex queue network in which two subsystems connected in a biserial way further linked with a common server. Chen [8] built up the participation capacities to examine the consistent state conduct of lining frameworks having differing bunch sizes. Creator utilized a nonlinear programming system with the combination of Zadeh's augmentation rule to grow such capacity. Gupta et al. [9] explored a broad examination of a queuing model comprising of multi-server associated in a biserial way.
Suhasini et al. [10] created a two-terminal couple queuing model to examine the queuing parameters. Uma and Manoj [11] played out an exhaustive investigation of single server bulk queuing model including three phases of heterogeneous assistance. A bulk waiting line framework with single help has been examined by Thangaraj and Rajendran [12]. Mittal and Gupta [13] 2601 ANALYSIS OF TRI-CUM BISERIAL BULK QUEUE MODEL developed a biseries bulk queuing model connected with a common server in a steady-state condition. Agrawal and Singh [14][15][16][17] performed detailed exploration to calculate the various queuing performance measures of some recently established tri-cum biserial based queuing models.
Numerous applications can be observed in which the developed model can be efficiently implemented. For instant, in the gaming-club, three sections Sra, Srb, and Src exist. These sections comprise various games activities that can be played in a team only. The minimum players in a team are two and can go up to any higher number. The team enters in any of the sections and can randomly move from one section to another. It is also possible that after entering in only one section, they exit from the section Srd. Various combinations of the team's movements are possible.
These types of gaming-clubs are widespread in metropolitan cities and malls. Therefore, such situations may arise when teams /Customers have to wait for a long time to get availed of the facility. This is really a very complex problem that can be effortlessly managed by the developed model.

MATHEMATICAL DESCRIPTION OF THE MODEL
In this queuing model, three servers are connected in parallel in tri cum biseries way, which are further linked with a common server in series. The queues associated with the servers Sra, Srb, Src, and Srd are Qa, Qb, Qc, and Qd, respectively. The customers entered the system with mean arrival A similar criterion will apply to those customers who entered in servers Srb and Src. After availing the service at server Srd the customer is permitted to exit the system. The pictorial representation of the considered problem is demonstrated in Figure 1.

SOLUTION METHODOLOGY
The steady-state governing differential-difference equation of the model can be written as  TRI-CUM BISERIAL BULK QUEUE MODEL   , , ,  , , ,  ,  , ,  , ,  ,   1,  1, ,  1, ,  1,   1, , ,  1 1, 1, , c d n n n n a n B n n n b n n B n n c n n n B n a ab n n n n a ac n n n n a ad n n n n b ba n n n n b bc n n n n P P P P ,, ,, ( , , , ) f z z z z is of (0/0) indeterminate form. Therefore, using L-Hospital rule, we get Again differentiating numerator and denominator of Eq. (3) separately w.r.t. 2 z by taking 2 Again differentiating numerator and denominator of Eq. (3) separately w.r.t. 3 z by taking 3 Again differentiating numerator and denominator of Eq. (3) separately w.r.t. 4 z by taking 4 On solving equations (4)- (7), we get the following values of traffic intensities of servers The solution (Joint Probability) of the model in steady-state is written as

PERFORMANCE MEASURES
Mean queue length (average number of customers)

VALIDATION STUDY
In this section, we consider some special cases by setting the value of an appropriate parameter to validate our result with existing models.

Case I:
If we assume

PARAMETRIC STUDY
The detailed description of the governing equation and solution methodology of the present model has been given in sections II and III. In section IV, various queuing performance measures have been given. In section V, some particular cases have been discussed by setting the value of an appropriate parameter to validate our result with existing models.
The details of various input parameters used to calculate the various queuing performance measures have been presented in Table 1.  Table 2

CONCLUSION
In the present study, a complex queuing model has been established with the help of moment generating function and other statistical tools to find the various queuing performance measures such as length of queues, fluctuation in queues, and average waiting time for customers. The legitimacy of the present queuing model is checked by thinking about particular cases. A broad parametric examination has been acquainted with show the suitability of the current solution methodology. This parametric examination can be helpful in different viable applications, for example, shopping complexes, sports centers, businesses, and so forth.