Write a simulation showing how the balanced allocation...

Write a simulation showing how the balanced allocation paradigm can improve performance for distributed queueing systems. Consider a bank of $n$ FIFO queues with a Poisson arrival stream of customers to the entire bank of rate $\lambda n$ per second, where $\lambda<1$. Upon entry a customer chooses a queue for service, and the service time for each customer is exponentially distributed with mean 1 second. You should compare two settings: (i) where each customer chooses a queue independently and uniformly at random from the $n$ queues for service; and (ii) where each customer chooses two queues independently and uniformly at random from the $n$ queues and waits at the queue with fewer customers, breaking ties randomly. Notice that the first setting is equivalent to having a bank of $n M / M / 1$ FIFO queues, each with Poisson arrivals of rate $\lambda<1$ per second. You may find the discussion in Exercise $8.26$ helpful in constructing your simulation. Your simulation should run for $t$ seconds, and it should return the average (over all customers that have completed service) of the time spent in the system as well as the average (over all customers that have arrived) of the number of customers found waiting in the queue they selected for service. You should present results for your simulations for $n=100$ and for $t=10,000$ seconds, with $\lambda=0.5,0.8,0.9$, and $0.99$.

Write a simulation showing how the balanced allocation paradigm can improve performance for distributed queueing systems. Consider a bank of $n$ FIFO queues with a Poisson arrival stream of customers to the entire bank of rate $\lambda n$ per second, where $\lambda<1$. Upon entry a customer chooses a queue for service, and the service time for each customer is exponentially distributed with mean 1 second. You should compare two settings: (i) where each customer chooses a queue independently and uniformly at random from the $n$ queues for service; and (ii) where each customer chooses two queues independently and uniformly at random from the $n$ queues and waits at the queue with fewer customers, breaking ties randomly. Notice that the first setting is equivalent to having a bank of $n M / M / 1$ FIFO queues, each with Poisson arrivals of rate $\lambda<1$ per second. You may find the discussion in Exercise $8.26$ helpful in constructing your simulation.

Your simulation should run for $t$ seconds, and it should return the average (over all customers that have completed service) of the time spent in the system as well as the average (over all customers that have arrived) of the number of customers found waiting in the queue they selected for service. You should present results for your simulations for $n=100$ and for $t=10,000$ seconds, with $\lambda=0.5,0.8,0.9$, and $0.99$.

Key Concepts

Discrete Event Simulation

Discrete event simulation is a computational approach used to model the operation of systems as a sequence of discrete events in time. In the context of queueing systems, it allows for assessing performance metrics like waiting time and response time under different load balancing strategies.

Distributed Queueing Systems

This concept involves a network of multiple queues or servers that work together to handle a stream of incoming tasks or customers. Performance improvements in such systems hinge on effective load balancing strategies to prevent bottlenecks and ensure resources are utilized efficiently.

Performance Metrics in Queueing Systems

Key performance metrics such as the average time a customer spends in the system and the average number of customers waiting in line are essential for evaluating and comparing queueing systems. These metrics provide insight into the efficiency of the service process and the effectiveness of load balancing interventions.

M/M/1 Queue

An M/M/1 queue is a basic model in queueing theory where arrivals occur according to a Poisson process and service times are exponentially distributed, with a single server handling the tasks. It serves as a fundamental building block for understanding more complex distributed queueing systems.

Poisson Arrival Process

A Poisson arrival process models the random occurrence of events, such as customer arrivals, at a constant average rate. This is particularly useful in queueing theory since it captures the memoryless property and randomness of arrivals over time, providing a solid basis for analyzing service systems.

Exponential Service Times

Exponential service times are used to model the duration a server takes to complete a service, where the service duration has a memoryless property. This facilitates tractable mathematical analysis in queueing systems, especially in the M/M/1 queue model.

Balanced Allocation Paradigm

This concept refers to the strategy of assigning incoming tasks or customers across multiple servers or queues in order to evenly distribute the workload. Using a method such as the 'power of two choices' can significantly reduce peak loads and improve overall system performance by selecting the least loaded among a randomly chosen subset of servers.

Transcript

00:01 We are looking at a problem here about regarding wait time, waiting in a line at a fast food service window.

00:13 They're saying here for this problem that we're supposing that customers are arriving at the fast food restaurant at a rate of nine people per hour.

00:26 So this function, f of x, is saying f of x is the average time in hours.

00:37 A customer will wait in line before being served.

00:43 So that's what the f of x stands for the output of that function.

00:47 The input of this function, x, is the average number of people served per hour.

00:55 So we are going to look at a graph of that f of x.

01:01 Notice i typed in into the graph and calculator, the function 9 divided by x times x minus 9, whenever you're dividing in the calculator, you've got to put a parentheses around, in this case, the product in the denominator.

01:19 So we get this graph that looks like this, and the question a says, when you're looking at the graph, we're only seeing values for x greater than nine.

01:31 Why is this function meaningless for values x less than nine? well, look at the function.

01:43 If x were less than nine, what would this, what would f of x be? well, nine is just nine, but if x is less than than x, because it's a number of people, it has to be a positive number, somewhere between, well, could be zero to, if it's less than nine, zero to eight people.

02:07 So that number, i'm assuming, is a positive number.

02:12 And x is, if it's less than nine, that number minus nine is going to result in a negative number.

02:21 So this factor in the denominator would be negative times a positive x, which would make our denominator a positive number times a negative number would be a negative.

02:34 9 divided by a negative number is a negative number.

02:38 What that means is if x were less than 9, f of x would be negative.

02:45 And remember what f of x is.

02:46 It's an average weight time, and time can't be negative.

02:50 So it doesn't make any sense for x to be less than nine.

02:55 It has to be greater than nine for that f of x to be positive.

03:02 So also if you, what we know about rational functions, we know that there'll be the values of x being nine and x being zero.

03:14 We'll make the denominator zero, which would give us a vertical asymptote.

03:20 So in this function, when x is 9, there is no output.

03:27 There's a vertical asymptote there.

03:30 Part b says suppose, well, before part b, it says suppose the average time to wait on a customer, or to serve a customer is five minutes...