Queuing model ,different type of queuing techniques are use to control the congestion

1. Queuing Analysis
Based on noted from Appendix A of
Stallings Operating System text
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2. 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

3. 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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4. 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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5. 02/05/16 5
Single server queue
w = mean # items waiting
Tw = mean waiting time
queue
arrivals
λ= arrival rate
server
Dispatching
discipline
departures
Ts = mean service time
ρ = utilization
r mean # items residing in the system
Tr = mean residence time

6. Multi-server /single queue
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queue
arrivals
λ= arrival rate
Dispatching
discipline
server0
server1
Servern-1
……….

7. Multi-server /Multiple queues
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server0
server1
Servern-1
……….
queue
arrivals queue
queue

8. 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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9. 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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10. 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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11. 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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12. 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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13. 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
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14. 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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