1 State behavior description There are two ways to get the equations Monitor the behavior of of customers in a system Subject to arrivals and departures First way: Kolmogorov approach That we studied last time Second way: rate diagram Key
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Presentation Transcript
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1 State behavior description There are two ways to get the equations
Monitor the behavior of # of customers in a system
Subject to arrivals and departures
First way: Kolmogorov approach
That we studied last time
Second way: rate diagram
Key driver of today’s lecture<br>
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2 Birth and death process: Kolmogorov approach N(t) = # of customers
at time t. λn arrivals
(births) departures
(deaths) μn<br>
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3 Differential equation: steady state analysis Limiting case<br>
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4 Complex example: 2 queues in tandem State space
(n1, n2)
n1 = # customers in the first queue
N2 = # customers in the second queue
P(n1, n2) ? λ1 μ1 n1 λ2 μ1 n1 p 1-p<br>
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5 Kolmogorov approach Think in terms of P(n1, n2)(t+dt)
In order to end up having (n1, n2) at time t+dt
Where do I need to be at time t
Moreover, what event would take place
To have n1 customers in queue 1, and n2 in queue 2 at t+dt t+dt t (n1, n2 ) (n1, n2 ) (n1+1, n2 ) (n1, n2+1) (n1-1, n2 ) (n1, n2 -1 ) (1 – (λ1 +μ1+λ2 +μ2)dt)
μ1dt (1-p)
μ2dt
λ1 dt
λ2 dt (n1+1, n2 -1 ) μ1.dt.p<br>
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6 Solution according to the classical approach Limitation of the classical approach
Unmanageable when the problem
Gets more and more complicated<br>
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7 Rate diagram: simple problem Classical approach (for the case of one queue)
Steady state analysis
Balance equation 0 1 2 3 ….. n …..<br>
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8 Rate of transition Rate of transition from n to n+1
Average # times the system moves from n to n+1
=> Average # of arrivals when we have n customers
=> λn n n+1<br>
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9 Rate diagram: complicated problem Consider rate diagram approach
For the case of the two queues .
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10 Balanced system If the process is in equilibrium
=> the average # times (per u.t.)
That the process enters a state n
Is equal to
The average # times
The process exits state n
These are called
Balance equations
Rate into a state = rate out of a
state<br>
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11 Reversible Markov process Solving the equations
These equations are called
Local balance equations
Balance specific flows
If a system satisfies these individual local equations
=> Reversible Markov process
=> it will have a product form solution<br>
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12 Z-transforms: generating functions If we have a sequence of numbers {f0,f1 ,f2 , …,fk ,..}
It is often desirable to compress it into a single function
This process of converting a sequence of numbers
Into a single function is called the z-transformation
The resultant function is called the z-transform of numbers
The z-transform of a sequence is defined as<br>
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13 Z-transform: application in queuing systems<br>
