Question

Consider a taxi station where taxis and customers arrive in accordance with Poisson processes with respective rates of one and two per minute. A taxi will wait no matter how many other taxis are present. However, an arriving customer that does not find a taxi waiting leaves. Find (a) the average number of taxis waiting, and (b) the proportion of arriving customers that get taxis.

Short answer

In conclusion, (a) The average number of waiting taxis is \( \frac{1}{3}\). (b) The proportion of arriving customers that get taxis is \( \frac{2}{3}\).

Step-by-step solution

Identify the birth-death process components

In this case, let's identify the birth and death rates of the queue at any state i: - Birth rate (λ): The arrival rate of customers, which is given as 2 customers per minute. - Death rate (μ): The arrival rate of taxi, which is given as 1 taxi per minute. Let's denote the probability that there are i taxis waiting as P(i).

Formulate balance equations for the steady state

Now we will write balance equations to ensure traffic conservation in and out of the states: 1. For state 0, the balance equation would be λP(0) = μP(1), since there are no taxis waiting. 2. For states i ≥ 1, the balance equation would be λP(i - 1) = μP(i), since taxis can arrive and depart.

Solve balance equations

We shall solve the balance equations to find P(i) for all states. 1. For state 0, the equation \(λP(0) = μP(1)\) becomes \(2P(0) = P(1)\). 2. For states i ≥ 1, the equation \( λP(i - 1) = μP(i)\) becomes \(2P(i - 1) = P(i)\). Now, notice that we have a recursive relation in P(i), so the probabilities for states i ≥ 1 can be found as follows: \(P(1) = 2P(0)\) \(P(2) = 2P(1) = 4P(0)\) \(P(3) = 2P(2) = 6P(0)\) \(P(i) = 2i P(0)\), for all i.

Normalize probabilities

We need to normalize the probabilities to ensure:\( \sum_{i = 0}^{\infty} P(i) = 1\). Summation formula for i = 0 to infinity, 1 = P(0) + \(2P(0)\) + \(4P(0)\) + \(6P(0)\) + ... = P(0)(1 + 2 + 4 + 6 + ...) The sum of the series can be represented as: \( \sum_{i = 1}^{\infty} 2iP(0) = 2P(0) \sum_{i = 1}^{\infty} i \) The sum of positive integers up to infinity is given by: \(\sum_{i = 1}^{\infty} i = \frac{1}{4}\) Now, we have: 1 = P(0)(1 + 2 * \(\frac{1}{4}\)) = P(0)\(\frac{3}{2}\) Thus, we find the probability for state 0 as, \( P(0) = \frac{2}{3}\).

Find the average number of waiting taxis (a)

The average number of waiting taxis (L) can be found as: \( L = \sum_{i = 1}^{\infty} iP(i)\) Using our summation relation from step 4, we get: \( L = \sum_{i = 1}^{\infty} i(2iP(0)) = 2P(0) \sum_{i = 1}^{\infty} i^2\) Knowing that the sum of the squares of positive integers up to infinity is given by: \(\sum_{i = 1}^{\infty} i^2 = \frac{1}{12}\) We can now calculate the average number of waiting taxis: \(L = 2P(0) * \frac{1}{12} = 2 * \frac{2}{3} * \frac{1}{12} = \frac{1}{3}\) So, the average number of waiting taxis is 1/3.

Find the proportion of arriving customers that get taxis (b)

The proportion of arriving customers who get taxis is given by the probability of not finding a waiting taxi (i.e., in state 0, which is P(0)): Proportion = P(0) = \(\frac{2}{3}\) So, 2/3 of the arriving customers get taxis. In conclusion, (a) The average number of waiting taxis is 1/3. (b) The proportion of arriving customers that get taxis is 2/3.

Key concepts

Birth-Death Process

The birth-death process is a special type of Markov process that models various real-world systems, including taxi queues, biological populations, and customer service. At its core, this process features 'births', which increase the quantity in the system, and 'deaths', which decrease it. Each state in the process represents the number of items, organisms, or customers present at any given time.

Underlying the dynamics of the birth-death process, we have two key rates: the birth rate (\(\lambda\)) and the death rate (\(\mu\)). In our taxi queue example, the birth rate represents the arrival of customers (2 per minute), and the death rate represents the arrival of taxis (1 per minute). The difference between these two rates hints at how the queue length will evolve over time.

When birth rates exceed death rates, the queue is expected to grow; conversely, when death rates surpass births, the queue shrinks. Achieving a balance between these rates is crucial for reaching a steady state, which is where the queue neither grows nor diminishes over time. The balance equations establish this harmony, by equating the rate of entering a state (births) to the rate of leaving it (deaths). Solving these equations, as shown in the textbook exercise, gives us the steady state probability for each number of taxis waiting at the station.

Queueing Theory

Queueing theory investigates the behavior of waiting lines, or queues. From customers waiting to be served at a bank to data packets queuing in network routers, the applications are widespread. This theory uses mathematical models to describe systems where 'servers' provide 'service' to 'customers', who arrive over time.

Considering a Poisson process, both the arrival of customers and services are random events distributed over time. The taxi station in our exercise exemplifies such a system: the taxis are the 'servers', the customers are the 'service recipients', and both arrivals are assumed to occur randomly and independently in time, following a Poisson distribution with given rates.

Queueing theory provides tools to answer pressing operational questions: What is the average wait time? How many servers should be available to minimize delays? The analysis in the exercise applies queueing theory to calculate both the average number of waiting taxis and the proportion of arriving customers that get taxis, offering insights into the efficiency of the taxi service system.

Steady State Probability

The steady state probability refers to the long-term stability condition of a system, where the probabilities of being in any given state do not change over time. In the context of a birth-death process within queueing theory, it describes the likelihood that there will be a certain number of taxis waiting at the station at any moment.

The key goal is to find a set of probabilities for each possible number of waiting taxis such that the overall system reaches a balance. This balance ensures that, over time, no net change occurs in the distribution of probabilities across different states, i.e., the queue reaches equilibrium.

The method used to obtain these probabilities involves creating a series of balance equations, as seen in the textbook exercise, and solving them under the normalization condition that all probabilities must add up to one. The results indicate the steady state operation of the taxi station, informing the service provider about the average operating conditions, and helping to make data-driven decisions to improve service efficiency.