Q6. Consider a taxi station where taxis and customers arrive in accordance with Poisson processes (i.e.,...

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Q6. Consider a taxi station where taxis and customers arrive in accordance with Poisson processes (i.e., waiting time for the next customer is an exponential random variable) with respective rates of one and two per minute. Each taxi only takes one customer. A taxi will wait no matter how many other taxis are present. However, an arriving customer that does not find a taxi waiting leaves. The arrival and departure of taxis can be modeled as an M/M/1 queue. (Hint: this is similar to Q5 of HW5. Now the departure rate is 2 per unit time (minute) and the arrival rate is 1.) 1. (5 pts) Formulate the balance equation. 2. (10 pts) Solve for P₁. (Hint: use the same procedure as the standard M/M/1 queue in Lecture 24) 3. (10 pts) Compute the average number of taxis waiting. 4. (5 pts) Compute the proportion of arriving customers that get taxis (Hint: the proportion of arriving customers that gets taxis is the proportion of arriving customers that find at least one taxi waiting. Equivalently, it is the probability that at least one taxi is waiting). Q6. Consider a taxi station where taxis and customers arrive in accordance with Poisson processes (i.e., waiting time for the next customer is an exponential random variable) with respective rates of one and two per minute. Each taxi only takes one customer. A taxi will wait no matter how many other taxis are present. However, an arriving customer that does not find a taxi waiting leaves. The arrival and departure of taxis can be modeled as an M/M/1 queue. (Hint: this is similar to Q5 of HW5. Now the departure rate is 2 per unit time (minute) and the arrival rate is 1.) 1. (5 pts) Formulate the balance equation. 2. (10 pts) Solve for P₁. (Hint: use the same procedure as the standard M/M/1 queue in Lecture 24) 3. (10 pts) Compute the average number of taxis waiting. 4. (5 pts) Compute the proportion of arriving customers that get taxis (Hint: the proportion of arriving customers that gets taxis is the proportion of arriving customers that find at least one taxi waiting. Equivalently, it is the probability that at least one taxi is waiting).

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The problem describes a system that can be modeled as an MM1 queue where taxis and customers arrive ... View the full answer