Question

In the Erlang loss system suppose the Poisson arrival rate is \(\lambda=2\), and suppose there are three servers, each of whom has a service distribution that is uniformly distributed over \((0,2)\). What proportion of potential customers is lost?

Short answer

The proportion of potential customers who are lost in this Erlang loss system is given by the Erlang B formula. With the given Poisson arrival rate λ = 2 and uniformly distributed service rate μ = 1 over 3 servers, the traffic intensity ρ = \(\frac{2}{3}\). The Erlang B formula yields a probability of all servers being busy (P_loss) of \(\frac{1}{2}\), or 50%.

Step-by-step solution

Identify relevant parameters

In this problem, we are given the Poisson arrival rate λ = 2, the number of servers n = 3, and the service distribution as uniform over the interval (0, 2).

Compute the average service rate

Since the service distribution is uniformly distributed over the interval (0, 2), we can obtain the average service rate (µ) as the midpoint, which is \(\frac{0+2}{2}\), giving us µ = 1.

Compute the traffic intensity (ρ)

Traffic intensity (ρ) is the ratio of arrival rate to the average service rate times the number of servers: \(ρ = \frac{λ}{n \cdot µ}\). We plug in the given values and compute the traffic intensity: ρ = \(\frac{2}{3 \cdot 1}\) = \(\frac{2}{3}\)

Calculate the probability of all servers being busy (P_loss)

Using the Erlang B formula for loss systems, we can find the probability of all servers being busy (P_loss) when a customer arrives. The Erlang B formula is given as: \[P_loss(E, n) = \frac{\frac{(E^n)}{n!}}{\sum_{k=0}^n \frac{(E^k)}{k!}}\] where \(E = ρ \cdot n\) is the offered load and n is the number of servers. Plug in the values: E = \(\frac{2}{3} \cdot 3\) = 2 Let's compute the numerator and denominator separately: Numerator = \(\frac{2^3}{3!} = \frac{8}{6}\) Denominator = \(\frac{1}{0!} + \frac{2^1}{1!} + \frac{2^2}{2!} + \frac{2^3}{3!} = 1 + 2 + 2 + \frac{8}{6}\) Now, compute P_loss: \[P_{loss} = \frac{\frac{8}{6}}{1 + 2 + 2 + \frac{8}{6}} = \frac{\frac{8}{6}}{5 + \frac{8}{6}} = \frac{8}{16}\] The proportion of potential customers who are lost is P_loss = \(\frac{1}{2}\), or 50%.