1. Modeling and simulation of systems M/M/n model Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. System M/M/n  If the arrival of customers into the system with one service canal exceeds the possibilities of service so , the problem springs up. There exist more possibilities for solving of such problem:  Increase the number of service canals  Limit the lenght of queue or arrival of customers  Shorten the period of service, it means increase the parameter   The first possibility leads to the systems with many service canals of the type M/M/n.

3. System M/M/n with unlimited lenght of queue 0 1 n+1 n nn  22 nn nn This system has n equal canals. The customers are waiting in the queue if all the canals are busy. The queue is common for all canals. Kolmogorov´s differential equations: For k  n: For k  n:

4. It is called the intensity of operation System M/M/n with unlimited lenght of queue k  n: k  n: Substitution is used for the solution Then is in force: k=1,2,..,n If We get the form: For the stable state is in force:

5. System M/M/n with unlimited lenght of queue For k = n+1,..,n+r For k  n is in force: From condition if is possible to derive:

6. System M/M/n with unlimited lenght of queue  The condition of stabilization of system is: Relation and  is called the use of service It is possible to prove that the mean value of number of busy canals OBS k =  The mean value N r of number of customers in queue: The average number of customers in teh system N s : N s = OBS k + N r

7. System M/M/n with unlimited lenght of queue The middle period of waiting in queue T r : The quantity  is called intensity of operation. From the condition of stabilization is n . It means that the smallest whole number bigger than  is the smallest number of canals needed for stabilization of system. The average number of free canals is n - . The meaning of usage of service is clear from the relation. The average number of arriving customers is in numerator and the average number of those customers that system is able to serve are in denominator

8. System M/M/n with the limited lenght of queue  The customer who comes into system when the lenght of queue is maximal leaves from system without service. There is the loss of customers in these systems. 0 1 n+r n nn  22 nn Kolmogorov´s differential equations: k nk n n  k  n+r

9. System M/M/n with limited lenght of queue k  n For the stable state is in force: n  k  n+r The sum of probabilities p 0 * + p 1 * +...+ p* n-1 means probability that the customer will be served withou waiting. The sum of probabilities p n * + p* n+1 +...+ p* n+r-1 means probability that the customer will wait. The probability p * n+r means that the customer leaves without service (loss of customer).

10. System M/M/n with limited lenght of queue The mean value of number of customers in the system: The mean value of number of customers in the queue: The average number of busy canals: The mean period of waiting in the queue:

11. System M/M/n with limited lenght of queue  When we determine the mean value of waiting in the queue it is necessary to think of the fact that if the customer comes at the moment when there are n+k-1 customers in the system, customer will wait on average k/n , if was not refused. It means that not with the probability p * n+k-1, but with the probability p * n+k-1 /(1- p * n+r )

12. System M/M/n without queue  Such system means that the customer is served immediately or leaves the system without service Kolmogorov´s equations for the stable state: We get from the equations: Then for p * k is in force :

13. System M/M/n without queue  The probability p n * means that the customer will not be serviced  The probability 1- p n * means that the customer will be serviced immediately  In this case the middle value of the number of customers equals to the middle value of the busy canals The penetrability of the system  (the middle value of the number of serviced customers). If arrival for time unit is of customers,  / is the probability that the customer was serviced