EE8405 Chap.7 Finite Buffers and State Dependence
Finite Buffers and State Dependence
Finite buffers close the state space and make the full state a blocking boundary. This chapter constructs stationary probabilities from adjacent birth-death ratios, normalises them, uses PASTA to identify blocking for Poisson arrivals and converts offered flow into admitted throughput before applying Little's formula. State-dependent rates are handled by a cumulative product rather than a constant geometric ratio.
What this chapter covers
- 01
Finite state spaces
- 02
Boundary transitions
- 03
Stationary normalisation
- 04
PASTA
- 05
Admitted throughput
- 06
State-dependent rates
Worked example · free
Solve an M/M/1/2 queue
- 2Build relative state weights.
- 3Normalise and identify the full-state probability.
- 3Use admitted throughput for delay and check flow balance.
Key terms
- Capacity State
- The full state in which an additional offered arrival cannot enter.
- PASTA
- The property that Poisson arrivals see time-average state probabilities.
Finite Buffers and State Dependence FAQ
Can a finite queue have a stationary distribution above unit offered load?
Yes. Its state space is finite and excess arrivals can be blocked, although high offered load may produce severe loss and near-continuous server use. Report that loss explicitly.
Which rate enters admitted-customer delay?
Use offered rate multiplied by one minus blocking. Blocked customers do not spend time within the service system. This admitted flow matches the delay population. Keep the boundary visible.
How are state-dependent probabilities built?
Multiply successive birth-to-death rate ratios to express every state relative to state zero, then normalise the complete finite set. Finally verify that the probabilities sum to one and adjacent flows balance.
Why can delay look low in a heavily loaded finite queue?
A full system rejects offered customers, and rejected customers contribute no time inside the service boundary. If delay is calculated only for admitted customers, severe blocking can keep the admitted population from growing without bound. Report blocking beside admitted delay so the apparent improvement is not mistaken for good overall service.
Use carried throughput in Little's formula and name the excluded customers explicitly.
What checks expose a reversed state-rate ratio?
At light load the empty state should dominate, while at very high offered load the full state should become more likely. Adjacent local flows should also balance under the stationary distribution. If probabilities move toward emptiness as arrival pressure rises, or one adjacent balance fails, inspect whether a birth and death rate were inverted.
Boundary arrows deserve special attention because departures vanish at zero and admissions vanish at full capacity.
Why should blocking and delay be reported together?
Admitted-customer delay can stay modest because excess demand is rejected. Pairing it with blocking reveals whether low delay was purchased by denying service. Add utilization for context and name the excluded population.
Exam move
Draw every boundary arrow and use flow balance between admitted arrivals and busy-server departures as an independent check on the stationary solution. Start by numbering the states from empty to full and place each birth or death rate on the arrow it controls. At state zero, departures are disabled; at state N, new offered arrivals are blocked.
These boundary changes are part of the model, not exceptions added after the equations. Express each stationary probability relative to state zero by multiplying successive birth-to-death rate ratios. Keep the normalising constant symbolic until every state has been included, then solve it from the requirement that probabilities sum to one. Check local flow across each adjacent pair with numerical values.
A failed equality identifies the transition copied incorrectly. Use limiting cases before performance measures: low offered traffic should concentrate probability near zero, while very high traffic should concentrate it near the full state. For Poisson arrivals, use PASTA to connect the time probability of the full state to the blocking probability seen by arrivals.
State the Poisson requirement rather than treating the equality as universal. Convert offered arrivals to admitted throughput by multiplying by one minus blocking. Only admitted customers belong in a delay calculation, so apply Little's formula with that carried rate and the mean number inside the system. Report blocking and admitted-customer delay together.
Low delay caused by rejecting much of the offered demand is not automatically good service. Add utilization or mean busy servers to show what capacity is doing. For state-dependent systems, rebuild the ratio at every transition instead of forcing a constant geometric form. Ask whether the rate changes on entry to a state or departure from it, and use arrow labels to keep that direction clear.
Practise one model where service accelerates with occupancy and another where arrivals slow near capacity, then explain how each change reshapes the distribution. On the handwritten page, record the product-form ratio, the normalisation step, PASTA's scope, admitted rate and the matched Little boundary. During assessment, these five prompts turn a long finite-buffer derivation into a sequence that can be checked independently.
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