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Queuing Analysis

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An example of a Queue

• Web server: handles requests in 1 msec

• If requests arrive at a constant rate of 1000 req/sec or less, everything works fine.

• In reality, arrival rate is not constant but varies.

• Suppose arrival rate is irregular with an average of 500 req/sec.

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Single Server Queue

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Example

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Multiserver Queue

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Difference with Multi Single Server Queue

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Parameters

• Theoretical maximum input rate that can be handled by the system is:• In practice: 70-90%.

ST1

max

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Basic Queuing Relationships

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Queuing Formulas

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Example

• Messages arrive at a switching center for a particular outgoing communication line in a Poisson manner with a mean arrival rate of 180 messages per hour. Message length is distributed exponentially with a mean length of 14,400 characters. Line speed is 9600 bps.

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Example (cont’d)

• What is the mean waiting time in the switching center?

mean message length = 14400 X 8 = 115200 bits

average service time = Ts = 115200 / 9600 = 12 sec

arrival rate = = 180 / 3600 = 0.05 message/sec

utilization = = 0.05 X 12 = 0.6

mean waiting time = T = 0.6 X 12 / (1-0.6) = 18 sec

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Example (cont’d)

• How many messages will be waiting in the switching center for transmission on the average?

messages waiting = = 0.6X0.6/(1-0.6)= 0.9 messages

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Self Similar Traffic

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Self Similarity

• The idea is that something looks the same when viewed from different degrees of “magnification” or different scales on a dimension, such as the time dimension.

• It’s a unifying concept underlying fractals, chaos, power laws, and a common attribute in many laws of nature and in phenomena in the world around us.

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Cantor

Each left portion in a step is a full replica of the preceding step

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Self Similarity of Ethernet Traffic

• Seminal paper by W. Leland et al published in 1993, examined Ethernet traffic between 1989 and 1992, gathering 4 sets of data, each lasting 20 to 40 hours, with a resolution of 20 microseconds.

• Paper shattered the illusion of Poison distribution being adequate for traffic analysis.

• Proved Ethernet traffic is self similar with a Hurst factor of H = 0.9

• 0 < H <1 ; the higher H, the more self similar the pattern

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Nature of self-similar traffic

• Burstiness: small variations over small time periods, big variations over big time periods (as seen in the figure of slide 23)– As a result of this: If the traffic averaged over longer

periods is plotted, one sees the same percentage of variation as when averaged over short time periods (see figure on slide 25, columns left and right)

– Note: In the case of Poisson traffic, the percentage variations decrease as the time period over which the traffic values are averaged increases (see middle column)

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Self Similar Traffic in Simulation

• A superposition of many Pareto-distributed ON-OFF sources can be used to generate self similar traffic.

• Pareto distribution is a heavy-tailed distribution: the tail decays much more slowly than the exponential distribution.

• Typical sample includes many small values and a few very large values (bursty).

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How Inaccurate Are Older Models?

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Why is the Internet traffic self-similar ?

• It took long time to understand why the Internet traffic is rather self-similar. It appears that the TCP protocol, which is currently used by most applications over the Internet, introduces this traffic property. The speed of data transmission of TCP is influenced by congestion control when a packet gets lost. Through this mechanism, the many independent TCP connections that run over the Internet become dependent on one another. The interaction is quite complex and involves the retransmission process after time-out. The net result is that the traffic becomes self-similar.

• Note that voice and video streaming does not use TCP. As these types of applications become more important over the Internet, it can be expected that the traffic will become less self-similar.

• It is to be noted that the arrival pattern of new sessions (e.g. TELNET sessions or Web server sessions) have been observed to follow a Poisson distribution.