1. Finite-Source Multiserver Queue with Preemptive Priorities Alexandre Brandwajn School of Engineering University of California, Santa Cruz alexb@cse.ucsc.edu

2. Plan l Motivation l System considered l Simple recurrent analysis l Alternative method l Classes on several priority levels l Conclusions

3. Motivation l Priority service n computer application l Finite number of request sources l Multiple servers l Arbitrary number of classes

4. System considered

5. Assumptions l M servers l c classes l exponentially distributed n idle times n service demands l 1 class / priority level

6. Assumptions l Class i, i = 1,…c n N i sources 1/ i mean idle time 1/  i mean service demand l Class 1 highest priority

7. Simple recurrent analysis

8. l One class at a time l state description (n i,l i ) n n i users of class i n l i servers unavailable l servers vanish  i (n i,l i ) reappear  i (n i,l i )

9. Simple recurrent analysis l Approximation  i (n i,l i )  i (l i )  i (n i,l i )  i (l i ) l two-dimensional birth & death n p(n i,l i ) l starting with class 1

10. Simple recurrent analysis

12. Results with 3 servers

13. Results with 2 servers

14. Parameter sets

15. Set 5 with 3 servers

16. Set 6 with 3 servers

17. Set 7 with 2 servers

18. Simple recurrent analysis l Generally within confidence intervals l occasionally more significant errors n more likely with longer service at higher priority n not systematic n growth with ratio of service times –peak and vanish

19. Alternative method

21. Alternative approach l Pair (i, i+1) n State description (n i,n i+1,l i ) n n i, n i+1 users of class i and i+1 n l i servers unavailable to pair

22. Alternative approach l Servers vanish  i (n i,l i ) reappear  i (n i,l i ) l Pair (1,2) keep l other (i,i+1) keep results for i+1

23. Set 6 with 3 servers

24. Set 7 with 2 servers