4' Models with General Arrival or

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4. Models with General Arrival or Service Pattern
4.1 M/G/1 4.2 M/G/c 4.3 G/M/1, G/M/c
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4.1 M/G/1

• General Service Time

b(t) the probability density function of
service time S
B(t) the cumulative distribution function of
service time S

• Poisson Arrival Process

X(t) the number in the system
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4.1 M/G/1
The service times of customers are assumed to be
independent.
Notably,
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4.1.1 The Pollaczek-Khintchine Formula
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4.1.1 The Pollaczek-Khintchine Formula
Since Poisson arrival sees time average, we have
L L(D).
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4.1.1 The Pollaczek-Khintchine Formula

• Example

Consider a M/M/1 queue with arrival rate of ?
10/hr and mean service time 5 min. The training
course will decreases the standard deviation of
service time from 5min to 4 min but increases
the mean service time from 5 min to 5.5 min.
Should they have further training?
Performance is much more sensitive to mean than
to its variation.
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4.1.2 Departure-Point S-S ?n

• Transition probabilities of M/G/1 Queue

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4.1.2 Departure-Point S-S ?n

• Example

Machines break down according to a Poisson
process with rate of ? 5/hr. The repairing
time is 9 min with probability 2/3 and 12 min
with probability 1/3. Find the probability that
more than three machines will be down at any
time?
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4.1.2 Departure-Point S-S ?n

• Example (Contd)

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4.1.3 Proof that ?n pn

• Proof

X(t) system size at time t An(t)
number of unit upward jumps from state
n occuring in (0, t) Dn(t) number of
unit downward jumps to state n
occuring in (0, t)
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4.1.4 Waiting Times

• Generalization of Littles Formula

Laplace-Stieltjes Transform
where W(s) is the LST of W(t).
where L(k) is the factorial k-th moment of the
system size and Wk is the regular k-th moment of
W(t).
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4.1.4 Waiting Times

• Waiting Time Distribution of M/G/1

Here, B(s) is the LST of the service time B(t).
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4.1.4 Waiting Times

• Waiting Time Distribution of M/G/1

where R(s) is the LST of the residual-service-ti
me R(t) with CDF
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4.1.4 Waiting Times

• Waiting Time Distribution of M/G/1

• Example

In the previous example, if it loses 5000 per
hour that a machine is down, and that an
additional penalty must be incurred because of
the possibility of an excessive number of
machines being down. It is decided to cost this
variability at 10000?(std deviation of delay).
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4.1.5 Busy Period Analysis
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4.1.5 Busy Period Analysis
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4.1.6 M/G/1/K Queueing Models

• Transition Matrix

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4.1.6 M/G/1/K Queueing Models

• Departure Point and Arrival Point Probabilities

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4.1.7 Some Additional Results

• M/G/1 with Balking

b probability that an arrival decides to
actuaaly join the system

• M/G/1 with Non-preemptive Priorities

M/M/1 priority model can be extended to
nonexponential service time system. But
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4.1.7 Some Additional Results

• Output Process of M/G/1

M/M/1 has a Poisson output process, but M/G/1 no.
C(t) CDF of interdeparture times of M/G/1 queue