You might have a problem usually anticipated in capacity planning with the tools of queuing theory.
We begin our analysis of queueing systems by understanding Little's Theorem. Little's theorem states that:
The average number of customers (N) can be determined from the following equation:
N = λT
Here lambda is the average customer arrival rate and T is the average service time for a customer.
Proof of this theorem can be obtained from any standard textbook on queueing theory. Here we will focus on an intuitive understanding of the result. Consider the example of a restaurant where the customer arrival rate (lambda) doubles but the customers still spend the same amount of time in the restaurant (T). This will double the number of customers in the restaurant (N). By the same logic if the customer arrival rate remains the same but the customers service time doubles, this will also double the total number of customers in the restaurant.
Queueing System Classification
With Little's Theorem, we have developed some basic understanding of a queueing system. To further our understanding we will have to dig deeper into characteristics of a queueing system that impact its performance. For example, queueing requirements of a restaurant will depend upon factors like:
* How do customers arrive in the restaurant? Are customer arrivals more during lunch and dinner time (a regular restaurant)? Or is the customer traffic more uniformly distributed (a cafe)?
* How much time do customers spend in the restaurant? Do customers typically leave the restaurant in a fixed amount of time? Does the customer service time vary with the type of customer?
* How many tables does the restaurant have for servicing customers?
The above three points correspond to the most important characteristics of a queueing system. They are explained below:
* The probability density distribution that determines the customer arrivals in the system.
* In a messaging system, this refers to the message arrival probability distribution.
* The probability density distribution that determines the customer service times in the system.
* In a messaging system, this refers to the message transmission time distribution. Since message transmission is directly proportional to the length of the message, this parameter indirectly refers to the message length distribution.
Number of Servers
* Number of servers available to service the customers.
* In a messaging system, this refers to the number of links between the source and destination nodes.
Based on the above characteristics, queueing systems can be classified by the following convention:
Where A is the arrival process, S is the service process and n is the number of servers. A and S are can be any of the following:
Exponential probability density
All customers have the same value
Any arbitrary probability distribution
From: Felipe Besson [mailto:[hidden email]]
Sent: Friday, January 23, 2015 4:12 PM
To: [hidden email] Cc: Daniel Cukier
Subject: Solr I/O increases over time