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

2. 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?
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B.ramamurthy

3. 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) ?
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B.ramamurthy

4. 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.
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B.ramamurthy

5. 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.
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B.ramamurthy

6. 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.
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B.ramamurthy

7. 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)
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B.ramamurthy

8. 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
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B.ramamurthy

9. 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

10. 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

11. 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

12. 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