1. Analysis of M/M/c/N Queuing System With Balking, Reneging and Synchronous Vacations Dequan Yue Department of Statistics, College of Sciences Yanshan University, China Wuyi Yue Department of Information Science and Systems Engineering Konan University, Japan

2. Outline  Introduction  System Model  Analysis 1. Steady-state equations 2. Block matrix solution method 3. Steady-state probabilities 4. Special cases  Conditional Distribution  Conclusions

3. Introduction  Practical background  Literature review  Purpose of our research

4. Practical Background  Balking and reneging - Customers who on arrival may not join the queue when there are many customers waiting ahead of them - Customers who join the queue may leave the queue without getting serviced - A common phenomenon in many practical queuing situations - Examples in communication systems, production and inventory system, air defence system and etc., see, Ancker and Gafarian (1962), Shawky(2000) and Ke (2007)

5. Practical Background (cont.)  Server vacations - In many real world queuing system, server may become unavailable for a random period of time -Server vacations can represent the time when the server is performing some secondary task -Single server queuing models with vacations have been studied by many researchers and applied in many fields such as communication systems, production and inventory system, computer networks and etc, see, Doshi(1986,1990), Takagi(1991) and Tian and Zhang(2006) -There are a few studies on multi-server vacation queuing system in the vacation model literature

6. Literature Review  Three types of the processes 1. Birth and death (BD) process 2. Quasi birth and death (QBD) process 3. Level-dependent quasi birth and death (LDQBD) process

7. Literature Review (cont.)  BD process model 1. M/M/c queue with balking and reneging (Montazer- Haghighi et.al. (1986)) -infinite buffer and a constant probability of balking, 2. M/M/c/N queue with balking and reneging (Abou-El-Ata and Hariri (1992)) -finite buffer and a state-dependent probability of balking, In these models, the state processes are classical BD processes. The steady-state probability have been obtained by solving the difference equations

8. Literature Review (cont.)  QBD process model 1. M/M/c queue with synchronous vacations (Tian et. al. (1999)) 2. M/M/c queue with synchronous vacations of partial server ( Zhang and Tian (2003)) In these models, the state processes are infinite QBD processes. The steady-state probability have been obtained by using matrix geometric solution method (Neuts (1981))

9. Literature Review (cont.)  LDQBD process model - M/M/c /N queue with balking, reneging and server breakdowns (Wang and Chang (2002)) - M/M/c /N queue with balking, reneging and synchronous vacation of partial servers (Yue et al. (2006)) In these models, the state process are finite LDQBD processes. The closed form expressions for steady-state probabilities have not been obtained

10. Purpose of Our Research  To study an M/M/c/N queue with balking, reneging and synchronous vacation  To present a closed form expression for computing the steady-state probability of the model  To present a block matrix method for solving a special queuing model with LDQBD process  To show that some existing models in the literature are special cases of our model

11. System Model  Customers arrive according to Poisson process with rate  Service time is assumed to be exponentially distributed with rate  A customer who on arrival finds customers in the system, either decides to enter the queue with probability or balks with probability  After joining the queue, a customer reneges if its waiting exceeds a certain time which is assumed to be exponentially distributed with rate  All the servers take synchronous vacation when the system is completely empty. The vacation time is assumed to be exponentially distributed with rate

12. Procedure of Analysis Step 1. Develop steady-state equations Step 2. Partition infinitesimal generator and steady-state probability vector Step 3. Rewrite steady-state equations in block matrix form Step 4. Find solution in block matrix form Step 5. Compute inversions of some block matrices Step 6. Find the closed form expressions of steady-state probabilities

13. Analysis  Notation the number of customers in system at time servers are taking on vacation at time servers are not taking on vacations at time is a Markov process with state space: Note: If we label the states in lexicographic order: then the Markov process is a level-dependent QBD Process

14. Analysis (cont.)

15. Steady-State Equations  Notation Steady-state probabilities:  Steady-state equations where is steady-state probability vector, is a generator, and is a column vector of order

16. Block Matrix Solution Method  Partitioned block structure where

17. Block Matrix Solution Method (cont.)  The steady-state equations in block matrix form where is a column vector of order with all its components equal to one

18. Solution in Block Matrix Form  Theorem 1. The segments of the steady-state probability vectors are given by where

19. Computing Inverse Matrix  Notations

20. Computing Inverse Matrix  Lemma 1-Inverse of matrix For the elements of the matrix is given by where are given by the following recursive relations where and

21. Computing Inverse Matrix (cont.)  Explicit expression of where is a row vector of order 2 and

22. Example 1

23. Computing Inverse Matrix (cont.)  Lemma 2- Inverse of matrix For the elements of matrix is given by the empty summation is defined to be zero.

24. Example 2

25. Steady-state Probabilities  Theorem 2 The steady-state probabilities are given by where

26. Special Cases  M/M/c/N queue with balking and reneging Let and then our model becomes the model studied by Abou-EI- Ataharir (1992)

27. Special Cases (cont.)  M/M/c queue with balking and reneging Let and then our model becomes the model studied by Haghighi et al. (1986)  M/M/c queue with synchronous vacation Let and then our model becomes the model studied by Tian et al. (1999)

28. Conditional Distribution  Let represent the conditional queue length given that all servers are busy, then we have Theorem 3 The conditional stationary distribution of the queue length is given by where and are given in Lemma 2 and Theorem 2

29. Conditional Distributions (cont.)  Remark 2. Based on Theorem 2, we can obtain some other performances such as the expected number of customers in the system and in the queue, etc. However, these performance have very complex expressions.

30. Conclusions  Studied an M/M/c/N queue with balking, reneging and synchronous vacations  Presented a block matrix method for solving the steady- state equations  Presented a closed form expressions for steady-state probabilities  Obtained some existing models in the literature as special cases of our model  Derived the conditional distributions for the queue length

31. Thank you very much!