Under otherwise the same assumptions as in exercise 9.22, it is assumed that a data record, which has been waiting in the buffer a random patience time, will be deleted as being no longer up to date. The patience times of all data records are assumed to be independent, exponential random variables with parameter \(v\). They are also independent of all arrival and processing times of the data records.

(1) Draw the transition graph.

(2) Determine the stationary loss probability.

Data from Exercise 9.22

A computer is connected to three terminals (for example, measuring devices). It can simultaneously evaluate data records from only two terminals. When the computer is processing two data records and in the meantime another data record has been produced, then this new data record has to wait in a buffer, when the buffer is empty. Otherwise the new data record is lost. The buffer can store only one data record. The data records are processed according to the FCFS-queueing discipline. The terminals produce data records independently according to a homogeneous Poisson process with intensity \(\lambda\). The processing times of data records from all terminals are independent, even if the computer is busy with two data records at the same time, and they have an exponential distribution with parameter \(\mu\). They are assumed to be independent of the input.

Let \(X(t)\) be the number of data records in computer and buffer at time \(t\).

(1) Verify that \(\{X(t), t \geq 0\}\) is a birth and death process, determine its transition rates and draw the transition graph.

(2) Determine the stationary loss probability, i.e., the probability that in the steady state a data record is lost.