Feedback of a Non-Truncated Erlang Queuing System with Balking and Retention of Reneged Customers

The aim of this paper is to derive the analytical solution of the non-truncated single-channel Erlangian queue: M/Ek/1 at the steady-state with adding the concepts of balking, feedback strategy and retention of reneged customers. We obtain the probabilities that there are n customers in the system and the customers in the service occupces stage s, (s = 1, 2, ..., k ), the probability of empty system and some measures of effecting of queuing system are obtained using the iterative method. Some important queueing models are derived as special cases of this system. Some numerical values are given showily the effect of correlation between the probabilities and the additional concepts.


Introduction
Has been studying the queue in which the service time Erlangian has recently been a good study. White et al. [1] studied non-truncated queue: M/E j /1 who obtained the solution from of generating function and the probabilities could be obtained by power series expansion. Kotb [2] derived an analytical solution of the state-dependent Erlangian queue: M/E j /1/N with balking. In 2005 Shawky [3] study the service Erlangian machine interference model: M/E r /1/k/N with balking and reneging. On a truncated Erlangian queueing system with state -dependent service rate, balking and reneging is obtained by Paoumy [7] using iterative method. Also, Madhu and Kumar [4] developed a M/E k /1 queueing system with working vacation. Other related studies are presented by Abid and Al -Madi [8], Kotb and Moamer [12], Mishra and Dinesh [6], Jayachitra and James Albert [10] and [11] and Jayachitra et al. [9]. Recently, Jeganathan et al. [13] Focused on M/E K /1/N queueing system with some important concepts.
In this article, we have proposed derive the analytical solution of the non-truncated single-channel Erlangian queue: M/E k /1 at the steady-state with adding the concepts of balking, feedback strategy and retention of reneged customers. The express probabilities that there are n customers in the system and the customer in the service occupies s stage, the probability of empty system are obtained using a recurrence relation. Some special cases are deduced, some measures of effectiveness. Finally, a simulation study has been considered to illustrate the numerical application for the model.

Basic Notations and Assumptions
To construct the system of this paper, we define the following parameters:   (2) Service time Erlangian queue having k-service stages each with rate n k µ µ = . The customers are served according to FIFO discipline.
(3) After completion of each service the customer either joins at the end of the original queue as a feedback customer with probability ( ) 1 q − or departure the system with probability q . (4) After joining the queue each customer will wait a certain length of time for service to begin with probability (1-p). If service has not begun by then, the customer will get impatient and leave the queue without getting service with probability ( )

Model Formulation and Analysis
From the above notations and assumptions and applying non-Markovian (Erlangian service) in the steady-state, we obtain the following probability differential-difference equations as: For solve the probability-difference equations above we use the iterative method as follows: Solving equation (1), we get: From the first equation of (2) and equation (5), we obtain: Substituting equation (4) into (6), one can easily get: From the second equation of (2) and equation (5), we get: Substituting equation (4) into (6), we obtain: Using the first equation of (3) with n=2 and equation (5), we find: Substituting equations (7) and (10) into (12), one gets: (13) it can be deduced by recursive manipulation that: As before, from the second equation of (3) at n=2 and equation (5), we get: Substituting equations (8) and (14) into (15), one gets: Also, from the first equation of (3) with n=3 and equation (5), we obtain: As before, Substituting equations (14) and (16) into (18) with 1, 2,3,..., ( 1) s k = − and by recursive manipulation, we get: As well, from the second equation of (3) with n=3 and equations (5) Also, from the first equation of (3) with n=3 and equations (5), (19) and (20) Using equations (11) And, from the above relationship, the expected number of units in the system and in the queue is:

Special Cases
Some queuing systems can be obtained as special cases of this system: Case (1): Let 0 α = and 1 q = , this is the non-truncated single-channel Erlangian queue: M/Ek/1 with balking studied by Kotb [2].

An Illustrative Example
Assume the parameters n = 2 units, k = 4 phase of service. The results of ns p , 0 , , ,  Solution of the model may be determined more readily by plotting 0 , , ns p p L and q L are drawn against , , , , q β λ µ α and p as given in Figures 1, 2, 3, 4, 5 and 6 respectively.       As we can see in graph1, shows that the increased the both of (arrival rate, service rate, Balking and reneging) and decrease the both of (the feedback strategy, and retention of reneged customers) offset it decrease the probability that there are no customers in the system. It is seen in graphs 2, 3, 4 and 5 respectively, shows that the increased the both of (arrival rate, service rate, Balking and reneging) and decrease the both of (the feedback strategy, and retention of reneged customers) offset it increase in the probability that there are n of customers in the system. Also in graph6, shows that the increased the both of (arrival rate, service rate, Balking and reneging) and decrease the both of (the feedback strategy, and retention of reneged customers) offset it increase in the expected number of customers in the system. And in graph7, shows that the increased the both of (arrival rate, service rate, Balking and reneging) and decrease the both of (the feedback strategy, and retention of reneged customers) offset it increase in the expected number of customers in the queue.

Conclusion
This paper has derive the analytical solution in steady-state for M/E k /1 with adding the feedback, balking and retention of reneged customers using the iterative method were devised to determine the probabilities that there are n customers in the system, the customer in service being in phase s and the probability that no customers are in the service department, the expected number of customers in the system and the expected number of customers in the queue. Finally, the numerical example was confirmed to confirm the model. The results, increasing (the probabilities that there are n customers in the system and the customer in service being in phase s, the number clients expected in the system) with increasing utilization factor, balking and retention of reneged customers. Also, the probability that there were no units in the system decreased with increasing utilization factor, balking and retention of reneged customers. Possible future extention of this work was include quality control process.