Queuing Analysis

Based on noted from Appendix A of Stallings Operating System text

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Queuing Model and Analysis• Queuing theory deals with modeling and analyzing systems with queues of items and servers that process the items.

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Queue1

Queue2

Queue3

server

Goals of Queuing Analysis

• Typically used in analysis of networking system; examples,– increase in disk access time– Increase in process load– Increase in rate of arrival of packets, processes

• Especially useful of analysis of performance when either the load on a system is expected to increase or a design change is contemplated.

• While it is a popular method in network analysis, it has gained popularity within a system esp. with the advent of multi-core processors.

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Analysis methods

• After the fact analysis: let the system run some n number times, collect the “real” data and analyze – problems?

• Predict some simple trends /projections based on experience – problems?

• Develop analytical model based on queuing theory – problems?

• Run simulation (not real systems) and collect data to analyze –problems?

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Single server queue

w = mean # items waitingTw = mean waiting time

queuearrivals

λ= arrival rateserver

Dispatching discipline

departures

Ts = mean service timeρ = utilization

r mean # items residing in the systemTr = mean residence time

Multi-server /single queue

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queuearrivals

λ= arrival rate

Dispatching discipline

server0

server1

Servern-1

……….

Multi-server /Multiple queues

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server0

server1

Servern-1

……….

queue

arrivals queue

queue

Parameters

• Items arrive at the facility at some average rate (items arriving per second) l.

• At any given time, a certain number of items will be waiting in the queue (zero or more);

• The average number waiting is w, and the mean time that an item must wait is Tw.

• The server handles incoming items with an average service time Ts;

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More parameters• Utilization, ρ, is the fraction of time that the

server is busy, measured over some interval of time.

• Finally, two parameters apply to the system as a whole.

• The average number of items resident in the system, including the item being served (if any) and the items waiting (if any), is r;

• and the average time that an item spends in the system, waiting and being served, is Tr; we refer to this as the mean residence time

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Analysis• As the arrival rate, which is the rate of traffic passing through the

system, increases, the utilization increases and with it, congestion. The queue becomes longer, increasing waiting time. At ρ = 1, the server becomes saturated, working 100% of the time.

• Thus, the theoretical maximum input rate that can be handled by the system is:

λmax = 1/Ts• However, queues become very large near system saturation,

growing without bound when ρ = 1. Practical considerations, such as response time requirements or buffer sizes, usually limit the input rate for a single server to 70-90% of the theoretical maximum.

• For multi server queue for N servers: λmax = N/Ts

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Specific Metrics• The fundamental task of a queuing analysis is as follows: Given the

following information as input:· Arrival rate· Service time• Provide as output information concerning:· Items waiting· Waiting time· Items in residence· Residence time.• We would like to know their average values (w, Tw, r, Tr) and the

respective variability the σ’s• We are also interested in some probabilities: what is probability

that items waiting in line < M is 0.99?

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Important Assumptions• The arrival rate obeys the Poisson distribution, which is equivalent

to saying that the inter-arrival times are exponential;• On other words, the arrivals occur randomly and independent of

one another.• A convenient notation has been developed for summarizing the

principal assumptions that are made in developing a queuing model.

• The notation is X/Y/N, where X refers to the distribution of the inter-arrival times, Y refers to the distribution of service times, and N refers to the number of servers.

• M/M/1 refers to a single-server queuing model with Poisson arrivals and exponential service times.

• M/G/1 and M/M/1 and M/D/1

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Little’s Law

• Very simple law that works from a Case Western Reserve University professor Dr. Little

• Average number of customers in a system = average arrival rate * average time spent in the system

• r = Tr * λ• w = Tw * λ• Tr = Tw + Ts• Extend it to the M/M/1 queuing model

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Examples

• Page 21-22-23• Database server (can be substituted for any

server).• Tightly-coupled multi-processor system• Necessary formulae are in pages: 14, 18 (Table

3 and Table 4)

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