EE8405 Chap.9 Queueing Networks and Mobile Systems
Queueing Networks and Mobile Systems
Networks create internal traffic as customers move and return between queues. This chapter solves flow-balance equations before applying node formulas, checks stability using effective rather than external rates and accumulates delay over repeated visits. Mobile systems add competing call and dwelling times, handover flow and fixed-point coupling between offered load and blocking.
What this chapter covers
- 01
Traffic equations
- 02
Routing probabilities
- 03
Feedback
- 04
Jackson networks
- 05
Channel holding time
- 06
Erlang fixed points
Worked example · free
Reveal feedback load in two queues
- 3Write one flow equation per node.
- 3Solve simultaneous effective rates.
- 2Check external flow balance and identify stability inputs.
Key terms
- Traffic Equation
- A balance equation for external and routed arrivals at a network node.
- Handover Probability
- The probability a mobile leaves a cell before its call ends and requires another channel.
Queueing Networks and Mobile Systems FAQ
Why not use external rates at each node?
Departures can be routed or fed back, so a node's effective load includes internal arrivals. Ignoring them can hide the true bottleneck and even miss instability.
How is channel holding time modelled?
With independent exponential call and dwelling times, channel holding is their minimum and has a rate equal to the sum of the two component rates.
What should a fixed-point solution report?
Show the update equations, initial guess, blocking calculation, stopping criterion and final relative change so another reader can reproduce convergence. Report the converged offered loads as well.
How do repeated visits change total network delay?
A customer contributes work and delay every time it visits a node, so summing one delay per node understates performance when feedback exists. Solve effective traffic first, then weight node delays by expected visit counts or reconcile total mean number across the network with external throughput through Little's formula.
State which route probabilities create the extra visits and verify that final external departures balance external arrivals.
What makes a mobile blocking calculation a fixed-point problem?
Handover traffic offered to one cell depends on calls admitted elsewhere, while admission itself depends on blocking produced by total offered load. The unknown loads and blocking probabilities therefore define each other. Begin from a declared guess, update consistently and stop only when a numerical tolerance is met.
Recompute all blocking values from the final loads and retain the last iterations so convergence is demonstrated rather than asserted.
Where is network stability checked?
Use each node's effective rate after routing and feedback, not its external input. One unstable node prevents a stationary network result even when average utilization looks low.
Exam move
Solve routing first, queue performance second. Check that external departures balance external arrivals and that every node is stable under its effective rate. Draw the network with external inputs, routing probabilities and final exits. Define the direction of every routing probability before using matrix notation. For each node, write effective arrivals as external flow plus routed departures from all upstream nodes.
Solve the simultaneous traffic equations and substitute the rates back into every equation. In an open stable network, total external input should equal total final departure. Calculate node utilization with effective rather than external rates. Feedback can make a node unstable even when its external load looks small, so check every node separately and stop stationary analysis if any utilization reaches one.
Once node performance is available, account for repeated visits. A customer returning to a queue contributes service demand and delay on every visit. Weight node delays by expected visit counts or reconcile total network number and external throughput through Little's formula. Practise a two-node feedback example by solving it both algebraically and through a visit-count argument.
For mobile systems, separate call completion and cell departure as competing clocks. The channel holding time is the minimum under the stated independence and exponential assumptions, so its rate is the sum of component rates. Handover probability follows from which clock rings first. Blocking at neighbouring cells changes handover flow, creating a fixed-point problem.
Document the initial guess, update equations, iteration order and stopping criterion. Track the maximum relative change, retain the last two iterations, and recompute all blocking values from final loads. Try a second starting guess when coupling is strong. On the handwritten page, include scalar traffic equations before the matrix form, node stability, visit weighting, competing-clock relations and a fixed-point checklist.
End a solution with the bottleneck node, the effect of feedback or handover, and the numerical evidence that the iteration has genuinely converged.
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