GOS vs blocking; 600 calls, 2.5 min

GoS vs blocking probability

• Grade of Service (GoS): the fraction of offered calls that are lost in the busy hour, $GoS = \dfrac{\text{calls lost}}{\text{calls offered}}$. It is a service-quality target or measurement for a route or the whole connection.
• Blocking probability ($P_B$): the probability, from a traffic model, that all servers are busy (time congestion) or that an arriving call is blocked (call congestion), e.g. from Erlang B.

For Poisson traffic in a loss system the two are numerically equal.

Given: calls offered $C_o = 600$, calls lost $C_l = 40$, mean holding time $h = 2.5\ \text{min} = \tfrac{1}{24}$ h, period T = 1 h.

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

(a) Traffic offered

$$\begin{aligned} A_o &= \frac{C_o\,h}{T} = 600 \times \tfrac{1}{24} \\ &= 25\ \text{E} \end{aligned}$$

(b) Traffic carried

Calls carried $= 600 - 40 = 560$

$$\begin{aligned} A_c &= 560 \times \tfrac{1}{24} = 23.33\ \text{E} \end{aligned}$$

(c) Traffic lost

$$\begin{aligned} A_l &= A_o - A_c = 40 \times \tfrac{1}{24} = 1.67\ \text{E} \end{aligned}$$

(d) Grade of service

$$\begin{aligned} GoS &= \frac{C_l}{C_o} = \frac{40}{600} = 0.0667 \end{aligned}$$

(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

$$\begin{aligned} T_c &= GoS \times 3600\ \text{s} = \frac{40}{600} \times 3600 \\ &= 240\ \text{s} \end{aligned}$$

Answer: offered traffic = 25 E, carried = 23.33 E, lost = 1.67 E, GoS = 0.0667, congestion duration = 240 s (4 min).