Question

The following problem arises in molecular biology. The surface of a bacterium consists of several sites at which foreign molecules-some acceptable and some not-become attached. We consider a particular site and assume that molecules arrive at the site according to a Poisson process with parameter \(\lambda\). Among these molecules a proportion \(\alpha\) is acceptable. Unacceptable molecules stay at the site for a length of time that is exponentially distributed with parameter \(\mu_{1}\), whereas an acceptable molecule remains at the site for an exponential time with rate \(\mu_{2}\). An arriving molecule will become attached only if the site is free of other molecules. What percentage of time is the site occupied with an acceptable (unacceptable) molecule?

Short answer

The percentage of time the site is occupied by an acceptable molecule is given by: Percentage of time occupied by acceptable molecule = \(100 \times \frac{λα\times \frac{1}{μ_2}}{λ \times ( \frac{α}{μ_2} + \frac{1-α}{μ_1})}\) And the percentage of time the site is occupied by an unacceptable molecule is given by: Percentage of time occupied by unacceptable molecule = \(100 \times \frac{λ(1-α)\times \frac{1}{μ_1}}{λ \times ( \frac{α}{μ_2} + \frac{1-α}{μ_1})}\) Use these formulas with given values of λ, α, μ₁, and μ₂ to calculate the percentages.

Step-by-step solution

Identifying the given parameters

We are given the following parameters: 1. Molecules arrive according to a Poisson process with parameter λ. 2. A proportion α of the arriving molecules is acceptable. 3. Unacceptable molecules stay for an exponentially distributed time with parameter μ₁. 4. Acceptable molecules stay for an exponentially distributed time with parameter μ₂.

Calculate probabilities related to arrivals

Given the proportion α of acceptable molecules, we can say that the probability of a molecule being acceptable upon arrival is P(A) = α. Therefore, the probability of a molecule being unacceptable is P(U) = 1 - α.

Calculate probabilities related to departure

Since the length of time an unacceptable molecule stays attached is exponentially distributed with parameter μ₁, we can say that the departure rate of an unacceptable molecule is λ_U = λ(1 - α)μ₁. Similarly, since the length of time an acceptable molecule stays attached is exponentially distributed with parameter μ₂, we can say that the departure rate of an acceptable molecule is λ_A = λαμ₂.

Calculate probabilities of site occupation

Using the provided information, we can now calculate the probabilities of the site being occupied by an acceptable molecule and an unacceptable molecule. In steady state, we can write: P(Occupied by acceptable molecule) = \(\frac{\text{Arrival rate of acceptable molecules} \times \text{average time occupied by acceptable molecule}}{\text{Total Arrival rate} \times (\text{average time occupied by acceptable molecule} + \text{average time occupied by unacceptable molecule})}\) P(Occupied by acceptable molecule) = \(\frac{λα\times \frac{1}{μ_2}}{λ \times ( \frac{α}{μ_2} + \frac{1-α}{μ_1})}\) Similarly, P(Occupied by unacceptable molecule) = \(\frac{λ(1-α)\times \frac{1}{μ_1}}{λ \times ( \frac{α}{μ_2} + \frac{1-α}{μ_1})}\)

Calculate the percentages of time the site is occupied

Now that we have probabilities for site occupation by an acceptable and unacceptable molecule, we can calculate the percentage of time the site is occupied by each type: Percentage of time occupied by acceptable molecule = \(100 \times P(\text{Occupied by acceptable molecule})\) Percentage of time occupied by unacceptable molecule = \(100 \times P(\text{Occupied by unacceptable molecule})\) Using values of λ, α, μ₁, and μ₂, these percentages can be calculated to determine the percentage of time the site is occupied by an acceptable and unacceptable molecule.