Traffic numerical: 800 calls, 50 lost

Definitions

• Blockage (blocking): the condition in which a call attempt cannot be connected because all suitable servers (trunks, switch paths) are busy; its probability is the blocking probability.
• Lost call: a blocked call in a loss system that is cleared and not served.
• Grade of Service (GoS): proportion of offered calls lost in the busy hour, $GoS = \dfrac{\text{calls lost}}{\text{calls offered}}$; a smaller value means better service.

Given: calls offered $C_o = 800$, calls lost $C_l = 50$, mean holding time $h = 3\ \text{min} = \tfrac{1}{20}$ h, period T = 1 h.

Traffic in Erlangs $= \dfrac{\text{calls} \times h}{T}$.

(a) Traffic offered

$$\begin{aligned} A_o &= \frac{C_o\,h}{T} = 800 \times \tfrac{1}{20} \\ &= 40\ \text{E} \end{aligned}$$

(b) Traffic carried

Calls carried $= 800 - 50 = 750$

$$\begin{aligned} A_c &= 750 \times \tfrac{1}{20} = 37.5\ \text{E} \end{aligned}$$

(c) Traffic lost

$$\begin{aligned} A_l &= A_o - A_c = 50 \times \tfrac{1}{20} = 2.5\ \text{E} \end{aligned}$$

(d) Grade of service

$$\begin{aligned} GoS &= \frac{C_l}{C_o} = \frac{50}{800} = 0.0625 \end{aligned}$$

(e) Duration of congestion

For pure-chance (Poisson) traffic the proportion of time all trunks are busy (time congestion) equals the proportion of calls lost (call congestion), so congestion time in the busy hour is

$$\begin{aligned} T_c &= GoS \times 3600\ \text{s} = \frac{50}{800} \times 3600 \\ &= 225\ \text{s} \end{aligned}$$

Answer: traffic offered = 40 E, carried = 37.5 E, lost = 2.5 E, GoS = 0.0625, congestion duration = 225 s (3.75 min).