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Chapter 5: Problem 43

Customers arrive at a two-server service station according to a Poisson process with rate \(\lambda .\) Whenever a new customer arrives, any customer that is in the system immediately departs. A new arrival enters service first with server 1 and then with server 2\. If the service times at the servers are independent exponentials with respective rates \(\mu_{1}\) and \(\mu_{2}\), what proportion of entering customers completes their service with server 2?

Short Answer

Expert verified
The proportion of customers completing their service with server 2 is given by the probability of no arrivals during the service time at server 1, which can be calculated as \(e^{-\lambda \frac{1}{\mu_1}}\).

Step by step solution

01

Analyze customer behavior and transitions

To start, we should consider a customer's possible outcomes when arriving at the service station: 1. The customer arrives and there is no other customer in the system, so they proceed to server 1, and then to server 2. 2. The customer arrives when another customer is in the system. In this case, the current customer immediately leaves the system, and the new arrival takes their place by starting service with server 1. Recall that we want to find the proportion of customers who complete their service at server 2. The only situation where a customer would fail to complete service with server 2 is if a new customer arrives while they are being served by server 1. If a customer can complete service with server 1 without being interrupted by another arrival, they will then move on to server 2 and complete their service.
02

Determine the probability that a customer is uninterrupted at server 1

We need to determine the probability that no customer arrives while a particular customer is being served by server 1. The service time for server 1 is exponentially distributed with rate \(\mu_1\), which means the average service time is \(\frac{1}{\mu_1}\). We want to find the probability that no arrival happens during the average service time of server 1. Since the arrival process follows a Poisson process with rate \(\lambda\), the probability of no arrival during an interval of length t is given by: \(P(\text{No arrivals in time interval }t) = e^{-\lambda t}\) In our case, \(t = \frac{1}{\mu_1}\). Plugging this into the formula, we have: \(P(\text{No arrivals during service time at server 1}) = e^{-\lambda \frac{1}{\mu_1}}\)
03

Obtain the proportion of customers completing service with server 2

The customers who complete service with server 2 are the ones who are not interrupted during their service time at server 1. Thus, the proportion of customers completing their service with server 2 is the probability of no arrivals during the service time at server 1: Proportion of customers completing service with server 2 = \(e^{-\lambda \frac{1}{\mu_1}}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Distribution
An exponential distribution is a continuous probability distribution. It is often used to model the time until an event occurs, such as the time between arrivals of customers to a service station.
In the context of the original exercise, the service times at the servers follow exponential distributions. This means the time a customer spends being served at server 1 or server 2 is variable but has an average value determined by the rates \(\mu_1\) and \(\mu_2\).
  • The rate of the exponential distribution \(\mu\) is the number of times an event occurs in a unit interval.
  • The probability of an event (in this case, a customer being served) happening within a specific time can be calculated using the exponential distribution's probability density function.
In server 1, the service rate is \(\mu_1\), meaning a customer spends on average \(\frac{1}{\mu_1}\) time units in service. This average concept is crucial to understanding how the system operates and aids in computing the probabilities of completing a service without interruptions.
Service Station
A service station in queueing theory is a location where customers come to receive services. In the problem provided, our service station consists of two servers. Each has unique characteristics defined by their service rates \(\mu_1\) and \(\mu_2\).
The functionality here is specific:
  • Incoming customers first go to server 1.
  • After being served by server 1, they proceed to server 2 if no further interruptions occur.
One unique aspect of this service station is the effect of new arrivals on current customers. If a new customer arrives while another is in the system, the current customer is "bumped" from the process, highlighting an unusual dynamic that is essential for understanding how this service station operates. This setup affects the analysis of how many customers complete their service as intended.
Queueing Theory
Queueing theory studies how entities line up for service and how resources are allocated to handle the demand. It informs us about the performance and behavior of systems, like our two-server service station.
One key application in this problem is analyzing how customers transition through the service station:
  • They must first be served at server 1.
  • If uninterrupted, they proceed to server 2.
  • The process's efficiency depends on arrival rates and service rates.
In essence, queueing theory helps us calculate vital metrics such as waiting times, queue lengths, and the probability of completing service without interruptions. In this exercise, queueing theory calculates the probability that no new customer arrives during a customer's service time at server 1, hence deciding their ability to proceed to server 2.
Probability Models
Probability models provide a mathematical structure to predict the behavior of systems under uncertainty. In the context of the provided problem, it models the continuous arrival of customers and the service mechanics at the station.
The Poisson process is a highly useful probability model that helps in predicting the arrival of events over a given time frame. Here, the arrival of customers follows a Poisson process with rate \(\lambda\):
  • This model helps predict probabilities such as "no arrival in the time service is ongoing,"
  • It serves to establish understanding about whether a customer will reach server 2.
The model, grounded in probability, thus offers a powerful way to predict and navigate the unpredictability of customer arrivals and service durations. It helps us facilitate strategic decision-making, anticipating customer flow and determining the likelihood of service completion.

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Most popular questions from this chapter

Suppose that \(\left[N_{0}(t), t \geqslant 0\right\\}\) is a Poisson process with rate \(\lambda=1\). Let \(\lambda(t)\) denote a nonnegative function of \(t\), and let $$ m(t)=\int_{0}^{t} \lambda(s) d s $$ Define \(N(t)\) by $$ N(t)=N_{0}(m(t)) $$ Argue that \(\\{N(t), t \geqslant 0\\}\) is a nonhomogeneous Poisson process with intensity function \(\lambda(t), t \geqslant 0\) Hint: Make use of the identity $$ m(t+h)-m(t)=m^{\prime}(t) h+o(h) $$

Consider a single server queuing system where customers arrive according to a Poisson process with rate \(\lambda\), service times are exponential with rate \(\mu\), and customers are served in the order of their arrival. Suppose that a customer arrives and finds \(n-1\) others in the system. Let \(X\) denote the number in the system at the moment that customer departs. Find the probability mass function of \(X\). Hint: Relate this to a negative binomial random variable.

An event independently occurs on each day with probability \(p .\) Let \(N(n)\) denote the total number of events that occur on the first \(n\) days, and let \(T_{r}\) denote the day on which the \(r\) th event occurs. (a) What is the distribution of \(\mathrm{N}(n)\) ? (b) What is the distribution of \(T_{1}\) ? (c) What is the distribution of \(T_{r}\) ? (d) Given that \(N(n)=r\), show that the set of \(r\) days on which events occurred has the same distribution as a random selection (without replacement) of \(r\) of the values \(1,2, \ldots, n\)

A cable car starts off with \(n\) riders. The times between successive stops of the car are independent exponential random variables with rate \(\lambda\). At each stop one rider gets off. This takes no time, and no additional riders get on. After a rider gets off the car, he or she walks home. Independently of all else, the walk takes an exponential time with rate \(\mu\). (a) What is the distribution of the time at which the last rider departs the car? (b) Suppose the last rider departs the car at time \(t\). What is the probability that all the other riders are home at that time?

Consider a two-server system in which a customer is served first by server 1, then by server 2, and then departs. The service times at server \(i\) are exponential random variables with rates \(\mu_{i}, i=1,2 .\) When you arrive, you find server 1 free and two customers at server 2 -customer \(\mathrm{A}\) in service and customer \(\mathrm{B}\) waiting in line. (a) Find \(P_{A}\), the probability that \(A\) is still in service when you move over to server 2 . (b) Find \(P_{B}\), the probability that \(B\) is still in the system when you move over to server 2 . (c) Find \(E[T]\), where \(T\) is the time that you spend in the system. Hint: Write $$ T=S_{1}+S_{2}+W_{A}+W_{B} $$ where \(S_{i}\) is your service time at server \(i, W_{A}\) is the amount of time you wait in queue while \(A\) is being served, and \(W_{B}\) is the amount of time you wait in queue while \(B\) is being served.
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