Mean Value Analysis: Operational Laws

The three laws used to evaluate system performance in queueing network system are

1. Utilization Law
2. Little's Law
3. Forced Flow Law

Utilization Law

During a period of time (T), tasks arrive at a server, the server processes them, and completed tasks leave the server. An observer of the server during (T) can identify

1. The number of tasks arriving at the server during (T), identified by (A)
2. The amount of time, during (T), that the server is busy processing tasks (B)
3. The number of tasks completed during (T), called (C)

These measurements result in the following calculations

1. The mean service time for a completed task, measured in a unit of time

   S = B/C

2. The output rate or throughput of the server, measured in tasks per unit of time

   X = C/T

3. The percentage of the observation period that the server is processing tasks, called the utilization of the server

   U = B/T

Multiplying mean service time by output rate results in the utilization of the server

B/C x C/T = B/T

The utilization law is

U(i) = X(i) x S(i) , where i identifies the server. [D4]

Little's Law

The total number of tasks processed by a server at each time interval (t) during an observation period (T) provides the total amount of waiting and processing time for all tasks during (T). The measurement of completed tasks (C), an observation period (T), as well as accumulated waiting time (W) during (T) lead to the following calculations.

1. The mean response time for a completed task, measured in a unit of time

   R = W/C

2. The mean number of tasks at a server

   n = W/T

3. The throughput of a server

   X = C/T

Multiplying mean response time by throughput results in the mean number of tasks at the server, or the mean queue length of the server.

W/C x C/T = W/T

Little's Law is the relation
n(i) = R(i) x X(i) , where i identifies the server. [D4]

Forced Flow Law

A transaction flows through a system of servers. A transaction may have several tasks completed by a server before it leaves the system. A server completes a task in the transaction each time the transaction is at the server.

During an observation period (T), an observer can obtain

1. The number of transactions completed by the system (C(0))
2. The number of tasks completed by server i, (C(i))

These measurements support the following calculations

1. The average number of tasks per transaction for server i, which is called the visit ratio of the server

   V(i) = C(i)/C(0)

2. The average number of transactions completed by the system during (T), which is the system throughput

   X(0) = C(0)/T

3. The average number of tasks completed by server i during (T), of the server throughput

   X(i) = C(i)/T

Multiplying the system throughput by the visit ration of a server gives the throughput of the server

C(i)/C(0) x C(0)/T = C(i)/T

The forced flow law is

X(i) = V(i) x X(0)

This law is important, because it shows that system throughput can be calculated with the knowledge of visit ratio and throughput of any one server in the system. In addition, knowing the visit ratio of all servers and the throughput of just one server, allows for calculation of the throughput of all servers in the system. [D4]

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