Multiplexing and Traffic Engineering¶

5-mark answer and 10-mark extension

For 5 marks - FDM, TDM and WDM principles¶

Multiplexing combines several independent tributaries for transmission over one higher-capacity link; a demultiplexer separates them at the destination. FDM translates each input onto a different carrier so all channels occupy non-overlapping frequency bands simultaneously. Bandpass filters and guard bands limit adjacent-channel interference. It suits radio, cable TV, satellite transponders, and analog carrier telephony.

TDM gives each input the full link bandwidth for a recurring time slot. Synchronous TDM reserves fixed slots, whereas statistical TDM labels and assigns slots only to active inputs. It suits PCM trunks such as E1/T1 and requires common clock and frame synchronization.

WDM is optical frequency multiplexing: lasers at distinct wavelengths \(\lambda_1,\lambda_2,\ldots\) are optically combined onto one fiber and separated by filters at the receiver. CWDM uses wider spacing and fewer channels; DWDM uses dense spacing for high-capacity long-haul systems.

Add for a 10-mark multiplexing comparison¶

The operating paths differ. In FDM, each source modulates a carrier, a summer forms the composite spectrum, and the receiver selects, demodulates, and low-pass-filters each band. If channel bandwidth is \(B_i\) and guard interval is \(G_i\), the link requires approximately \(B_T\ge\sum B_i+\sum G_i\). Nonlinear devices can create intermodulation products, so linear amplification and sharp filters matter.

In synchronous TDM, a commutator reads one sample or word from each tributary, adds framing/signaling where required, and transmits a serial frame; the receiving clock and frame detector drive the inverse commutator. For \(N\) tributaries each contributing \(b\) bits at \(f_s\) samples/s, ignoring overhead, \(R_b=Nbf_s\). Slots assigned to idle sources waste capacity, while statistical TDM improves utilization at the cost of addresses, buffering, and variable delay.

In WDM, optical sources must remain within wavelength tolerances; a MUX, fiber, EDFA/repeater, optional OADM/ROADM, and DEMUX form the path. Chromatic dispersion, nonlinear fiber effects, amplifier noise, and component cost limit reach/channel count, but each wavelength can transparently carry Ethernet, SDH, or another format.

Thus FDM is natural for continuous analog/RF channels, TDM for synchronized digital streams, and WDM for multiplying fiber capacity. Actual channel spacings, wavelength grids, and payload efficiency are system- and standard-dependent.

Practice target: 9 minutes for the 5-mark principles or 18 minutes for all paths, equations, diagrams, and the comparison table.

5-mark answer and 10-mark extension

For 5 marks - TDM and E1 core¶

Time-division multiplexing (TDM) shares one digital link by interleaving tributary words in time. A transmitter samples each source, inserts its bits into an assigned slot, and adds framing/signaling; a synchronized receiver identifies frame boundaries and routes each slot to the corresponding output. Synchronous TDM gives every source a fixed slot even when idle. Statistical TDM allocates labeled slots to active sources, increasing efficiency but introducing addressing, buffering, and variable delay.

Telephone speech is nominally low-pass limited to about \(3.4\,\text{kHz}\) and sampled at \(f_s=8\,\text{kHz}\). With 8-bit PCM,

European E1/PCM-30 interleaves 32 time slots, each 8 bits, once every \(125\,\mu\text{s}\); 30 slots can carry speech, TS0 carries framing/alarms, and TS16 is conventionally allocated to channel-associated signaling.

Add for a 10-mark E1/T1 extension¶

One E1 frame therefore contains

and its gross line rate is

TS1-TS15 and TS17-TS31 provide the 30 nominal \(64\,\text{kbit/s}\) bearer channels, totaling \(1.920\,\text{Mbit/s}\). TS0 carries frame alignment and service information. In the common PCM-30 channel-associated-signaling convention, TS16 carries signaling arranged over a 16-frame multiframe; other signaling arrangements, including common-channel signaling, are network-dependent, so “TS16 always carries CAS” is not universal. E1 line coding and detailed CRC/multiframe use also depend on the deployed recommendation.

The North American/Japanese T1/DS1 convention differs: each \(125\,\mu\text{s}\) frame has 24 eight-bit channel words plus one framing bit, hence \(24\times8+1=193\) bits and

Signaling may use robbed bits under traditional channel-associated arrangements. Therefore E1 is not merely T1 with a different name: slot count, framing, signaling, line rate, and regional standards differ.

Operationally, the PCM encoder anti-alias-filters, samples, compands and encodes each analog channel; the multiplexer reads one octet from each tributary, emits TS0 through TS31, and repeats at \(8\,\text{kHz}\). The receiver recovers clock, detects framing, demultiplexes slots, decodes PCM, and reconstructs speech. TDM provides deterministic channel rate and straightforward digital switching, but fixed idle slots consume capacity and slips/jitter or loss of frame alignment impair many channels together. E1 is used between exchanges, PBXs, radio base stations, and legacy access multiplexers.

Practice target: 9 minutes for the E1 core or 18 minutes for E1/T1 arithmetic, frame diagram, operation, and signaling caveat.

Exam-ready answer

Traffic intensity is the average simultaneous occupancy offered to a telecom resource. If calls arrive at mean rate \(\lambda\) calls per unit time and have mean holding time \(h\) in the same unit,

One Erlang means one circuit occupied continuously, for example one 60-minute call or thirty 2-minute calls during an hour. Offered traffic \(A_o\) is attempted demand; with blocking probability \(B\), carried traffic is approximately \(A_c=A_o(1-B)\) and lost traffic \(A_l=A_oB\). Grade of service (GoS) for a loss trunk group is commonly the busy-hour probability that an offered call is blocked.

For \(m\) identical circuits, Poisson arrivals, independent exponential holding times, full availability, and blocked calls cleared without a queue, Erlang-B gives

A numerically convenient recursion is

Example: for \(A=2\) Erlangs and \(m=4\) trunks,

About 9.52% of attempts are blocked, so this group fails a target GoS of 1% and needs more circuits or less offered load. Engineers invert the table/recursion to dimension PSTN or mobile trunk groups. Erlang-B is not a waiting model and does not predict queue delay; retrials, non-Poisson bursts, reserved circuits, and finite sources require another model or simulation.

Practice target: 8 minutes; define one Erlang, state all loss-system assumptions, write direct and recursive forms, and interpret the worked probability.

Exam-ready answer

Queuing theory models calls, packets, or users that arrive, wait if service is unavailable, receive service, and depart. A queue is specified by its arrival process, service-time distribution, number of parallel servers, system capacity, calling population, and discipline such as FIFO or priority. Kendall notation \(A/S/c/K/N/D\) records these features; M/M/1 means Poisson (Markovian) arrivals, exponential service, one server, conventionally infinite capacity/population and FIFO when omitted.

Grade of service (GoS) quantifies customer impairment. In a loss system it is commonly blocking probability; in a delay system it may be mean delay or \(P\{W_q>t\}\). It must therefore be stated with its metric and busy-hour condition. For any stable system with effective arrival rate \(\lambda\),

where \(L,W\) refer to the whole system and \(L_q,W_q\) to only the waiting line. Little's law is distribution-independent but requires consistent averages and a stable observation period.

For an M/M/1 server of rate \(\mu\),

Example: a router with \(\lambda=80\) packets/s and \(\mu=100\) packets/s has \(\rho=0.8\), \(L=4\) packets and \(W=0.05\) s; \(L=\lambda W=4\) verifies Little's law. As utilization approaches unity, delay rises sharply, so capacity needs headroom. Unlike Erlang-B, which clears a blocked call and measures loss, a router buffer admits waiting and trades lower loss for delay; finite buffers, bursty traffic, priorities, and non-exponential service require models beyond M/M/1.

Practice target: 8 minutes; define the queue and GoS metric, write Little's law and M/M/1 stability/results, then interpret one numerical example.