We consider a new queuing model with sequential two stations (stages), single server at each station, where no queue is allowed at station 2 and with no restriction at station 1. There is a FCFS service discipline in which the input stream is Poisson having rate
New models of queuing theory have been needed lately concerning the developments in areas such as production line, communication, and computer systems. One of these necessary models is for a tandem queuing system. Many important studies have been done in this area. The mean waiting time and the mean customer number at two tandem channels (servers) which are Poisson arrival and exponential service time were given in [
In real life, while there is waiting case in the service systems, because of obligations and urgency and unavailability of desired features, the loss may occur in the beginning of second stage as in our proposed model. Therefore, we have decided to construct such a model and analyze it. The following two examples can be proposed as potential applications of our model under some conditions.
The following examples can be proposed as the applications of our model on some topics.
(a)
(b)
More generally, the server transfers voice messaging to the recipient. In this system, the server is first service, the recipient is second service. If the recipient is busy, and then the server destroys voice messaging at that moment.
Let
For stage
The probability
We get the probability
The customer’s loss probability is given as
Let
If sum of two service rates
We will prove the theorem by using the following inequality:
Let
The mean number of customers in the system is optimized from Theorem
If the random variables
The joint probability mass functions of
The marginal probability mass function of
The marginal probability mass function of
The random arrivals and service times were generated from exponential distribution as seconds by using MATLAB 7.10.0 (R2010a) programming for this proposed model. The number of customers taken was 10000 and was performed in three iterations. Performance measures are calculated for different values of
For
Iteration number  Simulation results  Exact results  Optimal results  








100  2.8163  0.8452  2.7692  0.8308  2.2222  0.6667 
1000  2.8111  0.8433  2.7692  0.8308  2.2222  0.6667 
5000  2.8124  0.8436  2.7692  0.8308  2.2222  0.6667 
For
Iteration number  Simulation results  Exact results  Optimal results  








100  2.2661  0.2263  2.2500  0.2250  2.2220  0.2222 
1000  2.2599  0.2261  2.2500  0.2250  2.2220  0.2222 
5000  2.2500  0.2260  2.2500  0.2250  2.2220  0.2222 
For
Iteration number  Simulation results  Exact results  Optimal results  








100  3.1042  0.0311  3.1034  0.0310  3.0769  0.0308 
1000  3.1034  0.0310  3.1034  0.0310  3.0769  0.0308 
5000  3.1043  0.0310  3.1034  0.0310  3.0769  0.0308 
A new queuing discipline is given for a Markov model which consists of two consecutive channels and no waiting line between channels. In this model, steadystate equations, the mean sojourn time, the mean number of customers, and loss probability are obtained. Additionally, two theorems are given which are about optimization of performance measures and the independent of the number of customers, respectively. Performance measures are calculated for different values of
For further research, a model, in which a customer who completed his service in channel 1, blocks channel 1 with probability