B.Ramamurthy Appendix A Queuing Theory B.Ramamurthy Appendix A 11/9/2018 B.ramamurthy

Problem Consider the following problem. Message packets are sent from a computers on a LAN to systems on other networks through a router. Given the network specification and request pattern, determine queuing time at router, and queue length, say, 95% of the time and similar metrics. What information do you need? How will you model the network? What formulas or expressions will you use in your computation of metrics? What metrics? 11/9/2018 B.ramamurthy

Some More Questions What happens to file retrieval time when disk utilization goes up? How does response time change if both processor speed and number of users on the system are doubled? How many lines should a dial-in facility of a time-sharing system have? How many lines are needed on on-line inquiry center (call center) ? 11/9/2018 B.ramamurthy

Queuing Analysis Queuing analysis is one of the most important tools for answering such questions. The number of questions addressed that can be addressed by queuing analysis is endless and it touches every area we discussed in the operating systems course and also in networking. We will look at some practical application of queuing analysis and simple rudimentary formulas. 11/9/2018 B.ramamurthy

Alternatives Case Study: Scaling up a LAN to two buildings. Actual implementation and evaluation of metrics. Make a simple projection using prior experience. Computer simulate a model. Develop an analytic model based on queuing theory. 11/9/2018 B.ramamurthy

Example Disk that is capable of transferring 1000 blocks per second is considered as having 50% (1/2 the load) when it is transferring at 500 blocks per second. Response time is the time it takes to retransmit any incoming block. See graph. 11/9/2018 B.ramamurthy

Queuing Models Single server queue Entities: server, clients, queue (line) Parameters: arrival distribution, arrival times, service time, queue length, waiting time, server utilization (busy time / total time) 11/9/2018 B.ramamurthy

Queuing Parameters λ – average arrival rate (of requests) w – average queue length Tw – average wait time Ts – average service time Tq – average time spent in the system (Tw+ Ts) λmax = 1 / Ts theoretical maximum rate that can be handled by the system. See Table A.1 11/9/2018 B.ramamurthy

Queuing structure Server Tw – waiting time q – items in queuing system arrivals queue dispatch discipline departures W- items waiting Tw – waiting time q – items in queuing system Tq – queuing time 11/9/2018 B.ramamurthy

Multiple servers Multi-server , single queue Multi-server, multiple queue Theoretical maximum input rate: λmax = N / Ts for N servers Lets look at basic queuing relationships specified a well Little’s formulas. 11/9/2018 B.ramamurthy

Little’s Formula General Single server Multi-server q = lTq r = lTs r = lTs/N w = lTw q = w + r u = rN Tq = Tw + Ts q = w + Nr q -- mean number of items in the system – utilization : fraction of busy time u -- traffic intensity w – mean number of items waiting to be served l -- arrival rate 11/9/2018 B.ramamurthy

Queuing Models We would like to estimate (w,Tw, q,Tq), waiting items, waiting time, queued items, and queuing time, mean and standard deviation of each. Assumptions made about queuing model is summarized using a notation: G – general distribution, M – exponential distribution, D – deterministic fixed rate Accordingly models are named: M/M/1 refers to Poisson arrival , exponential service with a single server. 11/9/2018 B.ramamurthy