Erlang b

CIRCUIT SWITCHING MODEL

The PSTN (Public Switching Telephone Network) is a circuit switching network, which is characterized by the following features:
* End-to-end circuit (bandwidth) is reserved for each call
* Circuit switching is based on reserved bandwidth
* Other calls cannot use circuit
* Calls consist of three phases:
* Set-up phase: signaling messages requestand reserve bandwidth
* access signaling: dual tone multifrequency (DTMF)
* interoffice signaling: common channel signaling (SS7)
* Call is held: no delay except propagation; no processing in network
* Call termination (release) phase: signaling releases bandwidth
* Two types of switches:
* Analog telephone network uses spatial switches
* Currentdigital telephone network uses temporal switches
* Call blocking protects network from overload but telephone network can become overloaded by congestion
* problem: call requests propagate through switches, tying up links
* if unsuccessful, intermediate links will be cleared eventually but wasted in meantime
* increases probability that other calls will be blocked* those unsuccessful calls result in more waste
* failed calls will be retried, increasing the offered load
* congestion tends to increase itself: carried traffic is reduced while offered load increases
* some control mechanisms
* selective alternate routing and dynamic routing to less congested network areas
* call requests likely to failare blocked early at first switch
* code blocking: calls to congested area are blocked

Analog Telephone Network

In the old analog telephone network, a circuit is physical path for electrical analog 4-kHz signal.
* analog space-division switching
* Initially, only one switch was used
* NxN matrix (ie, N2 crosspoints)

* In 1953, Charles Clos formalized a type ofmultistage switching networks (Clos networks):
* multistage interconnection network (usually 3 stages) of crossbar modules
* each call entering an ingress crossbar switch can be routed through any of the available middle stage crossbar switches, to the relevant egress crossbar switch

* If k ≥ n, the Clos network is rearrangeably nonblocking, meaning that an unused input on aningress switch can always be connected to an unused output on an egress switch, but for this to take place, existing calls may have to be rearranged by assigning them to different centre stage switches in the Clos network.
* If k ≥ 2n - 1, the Clos network is strict-sense nonblocking, meaning that an unused input on an ingress switch can always be connected to an unused output on an egressswitch, without having to re-arrange existing calls.
* Number of crosspoints:

Digital Telephone Network

In current telephone networks, the voice signal is digitized and transmitted using time-division multiplexing:
* Digital switches transfer periodic time slots on input port x to output port y
* Reserved bandwidth is time slot instead of physical circuit
* Basic buildingblock is time slot interchange (TSI)
* modern digital switches are built from stages of time (T) and space (S) switches eg, AT&T No. 5 ESS is example of TST

Telephone Traffic
Assumptions:

* NxN telephone switch can be modeled as N independent servers
* Sources are identical and independent of each other and state of system
* Requests arrive in a single combined stream* In small time interval Δ, probability of arriving request is proportional to Δ
* Pr(arrival of request in Δ) = λΔ
* Pr(no request in Δ) = 1 - λΔ
* as Δ becomes very small, arrival stream is Poisson(λ)
* Holding times (service times) are exponential(μ)
* Traffic intensity (offered load) = (average arrival rate) * (average holding time)
* Erlang is the...