EE8405 Chap.6 Erlang Loss and Delay Systems
Erlang Loss and Delay Systems
Multi-server systems divide into loss and delay models according to what happens when all servers are busy. Erlang B applies when the arrival is cleared; Erlang C applies when it waits. The chapter tracks offered, blocked and carried traffic, uses a stable recursion for blocking, proves minimal capacity by checking the boundary and distinguishes probability of delay from mean waiting.
What this chapter covers
- 01
M/M/k/k loss
- 02
M/M/k delay
- 03
Erlang B recursion
- 04
Erlang C probability
- 05
Traffic accounting
- 06
Capacity dimensioning
Worked example · free
Dimension circuits by Erlang recursion
- 3Run Erlang B recursion in server order.
- 2Find the first capacity meeting the limit.
- 3Show the preceding capacity fails and interpret carried traffic.
Key terms
- Erlang B
- The blocking formula for a Poisson multi-server loss system without waiting.
- Erlang C
- The probability of waiting in a stable Poisson multi-server delay system.
Erlang Loss and Delay Systems FAQ
What is the quickest B versus C test?
Ask what happens when every server is busy. Immediate rejection selects Erlang B; retention in a queue selects Erlang C. The customer fate decides the model.
How is minimal capacity proved?
Show that the chosen server count satisfies the target and the immediately smaller count fails it. One successful value alone does not establish minimality. Keep both comparisons visible.
Does Erlang B insensitivity cover arrivals?
No. Its robustness concerns the holding-time distribution under the model conditions. Non-Poisson offered traffic can require different or approximate treatment. Arrival assumptions still need justification.
How should I organise an Erlang B recursion?
Use one row per server count and show the preceding blocking value, offered traffic, numerator, denominator and updated probability. Begin at the zero-server boundary and keep unrounded values through the target comparison. Add carried traffic to the chosen row as a physical check that mean busy servers do not exceed installed capacity.
A transparent table makes a copied rate or early rounding error easier to locate than a single factorial expression.
Why are probability of delay and waiting time separate targets?
Erlang C first identifies how often an arrival finds every server busy and must wait. The duration of that wait depends on the conditional waiting-time relation and spare aggregate service capacity. A system may create infrequent but long waits or frequent but short waits, so one percentage cannot describe both dimensions.
Read the service objective literally, calculate the named metric and state whether the result concerns any wait, mean wait or a tail probability.
What should the final capacity sentence contain?
Name the selected server count, offered traffic, service target, achieved metric and the failed smaller design, then state the arrival and holding assumptions. This makes minimality auditable.
Exam move
Build a recursion table and attach traffic accounting to every row. Always label whether a percentage is blocking, probability of delay or a waiting-time target. Begin every multi-server problem with the customer fate at full occupancy. If an arrival is cleared, use a loss model; if it remains and waits, use a delay model. Write that decision before selecting Erlang B or Erlang C.
Convert arrival rate and mean holding time to one unit, multiply them to obtain offered traffic in erlangs, and compare the load with the number of servers. For Erlang B, create a table with server count, previous blocking value, numerator, denominator and updated blocking value. Start from the zero-server boundary and retain enough digits for target comparisons.
The recursion is more auditable than a single factorial expression and adapts naturally to minimal-capacity searches. Prove minimality on both sides: the chosen count meets the blocking target and the immediately smaller count fails. Show both rows and keep the target unrounded. Then calculate carried traffic as offered traffic multiplied by one minus blocking and confirm that it does not exceed installed servers.
For Erlang C, check stability before evaluating delay. Separate probability of waiting from the duration of waiting, because a service objective may constrain either or both. If a tail target is requested, first obtain the probability of delay and then apply the conditional waiting-time relation with the correct spare-capacity rate. Revise insensitivity with precise boundaries.
The loss formula can be insensitive to the holding-time distribution under its model conditions, but this does not remove the Poisson traffic assumption and does not transfer automatically to a waiting queue. Practise explaining exactly which feature has dropped out. Add a sensitivity exercise in which offered traffic rises while capacity remains fixed.
Record the server count range that still meets the target and identify whether uncertainty comes from arrival rate or holding time. Close every design answer with the selected capacity, the metric and threshold it satisfies, the failed smaller design, and the assumptions under which the conclusion holds. On the formula page, keep loss and delay results in separate boxes so blocking is never reported as waiting probability.
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