^{1}

^{1}

^{1}

In order to enhance the precision of network simulations, the paper proposes an approach to adaptively decide the maximum of random variables that create the discrete probabilities to generate nodal traffic on simulated networks.
In this paper, a statistical model is first suggested to manifest the bound of statistical errors. Then, according to the minimum probability that generates nodal traffic, a formula is proposed to decide the maximum. In the formula, a precision parameter is used to present the degree of simulative accuracy. Meanwhile, the maximum adaptively varies with the traffic distribution among nodes because the decision depends on the minimum probability generating nodal traffic. In order to verify the effect of the adaptive maximum on simulative precision, an optical network is introduced. After simulating the optical network, the theoretic average waiting time of nodes on the optical network is exploited to validate the exactness of the simulation. The proposed formula deciding the adaptive maximum can be generally exploited in the simulations of various networks. Based on the precision parameter

Simulations are an important technique for the design of systems,
the estimation of performance, and the maintenance of systems [

For example, DQDB
networks are systems with asynchronous
transfer mode (ATM) and time-division
multiple access (TDMA) [

In simulations, random variables are used to create various
distribution functions of probabilities. These probabilities are applied to direct the input
amplitude of signals or noise as simulating communication systems [

In order to take precise simulations, this paper suggests a statistical model. Based
on the model, it is obvious that simulative
precision only depends on the maximum of random variables controlling the generation of network traffic. Based on the perception, a simple formula is proposed
to decide the feasible lower bound of maximums
of random variables. In the formula, the
feasible lower bound is dependent on a precision parameter, denoted by

So as to understand the
effect of the adaptive maximum on simulative precision, an optical TDMA network
is introduced. The MAC protocol of the optical TDMA network implements traffic
control. The average waiting time of a node on the network is in inverse
proportion to the traffic of the node [

In Section

Before simulations, a set of continuous probabilities is assigned to predefine the distribution of nodal traffic. In simulations, a set of discrete probabilities, which corresponds to the set of continuous probabilities, controls the generation of nodal traffic. The difference between continuous and discrete probabilities is used to manifest the influence of the maximum of integral random variables on simulative precision. Then the minimum probability in the set of continuous probabilities is exploited to decide the adaptive maximum of integral random variables.

In simulated networks, every node has one queue. Queues consist of cell (packet) buffers. Queues provide first-in-first-out (FIFO) service. The first cell buffer in a queue is attached to the transmission system of the simulated network. When an available slot on the transmission system is passed through, the contents of the first cell buffer will be written into the available slot.

The number of cells temporarily storing in queues is dependent on
the traffic generated by nodes. The heavier the nodal traffic, the longer the queuing delay. According to the complex MAC protocol of most networks,
an available slot on transmission
systems appears for some node
randomly. Hence, the prediction of the queuing delay of a specified cell is difficult. In order to
estimate performance, most simulations assume that networks are with heavy load [

For a node within
networks, every cell generated by disassembling procedures is first
stored in queues. Before the cell is transferred to transmission
systems, it must be sequentially shifted
into the first cell buffer
of nodal queues. When the cell is within the first cell buffer, the MAC protocol will be exploited to decide the moment after that
the node can write the cell out. In above operations, there are two moments relative to the queuing delay. The first moment is the instant
that a cell enters queues. The
second one is the flash that
the cell is sent onto transmission
systems. Let

Let

Let

The random variable

From (

If a simulation is with a precise-probability set

After discussing the enhancement of statistical precision,
statistics of the queuing delay are presented. According to (

If

Let

In order to guarantee
that the precise degree is acceptable,

Then,

In (

Because the recursive
formula of the power-residue method is computationally very efficient [

Then, a prime number
that is slightly greater than

In order for comprehending the effect of adaptive maximums of random variables on the precision of network simulations, an optical TDMA networks is introduced. Before depicting the structure of the network, the deduction for the average waiting time of nodes on the network (the waiting mean) is presented. The waiting time of a cell is the queuing delay that the cell waits in the first cell buffer of queues for an available slot on transmission systems. Based on the structure of the network, several working conditions are assumed for simulations. Due to these working conditions, an MAC protocol implementing traffic control is described.

For TDMA networks, a node must send requests to
preserve empty slots when it is going to transmit messages. More requests
preserve more slots. As the number of preserved slots of a node becomes large,
the waiting mean of the node will be reduced. Therefore, if a node has more
traffic, its waiting mean will be decreased. The relationship between the waiting
mean and the traffic of the

Because optical TDMA
networks are high-speed networks, the slot rate can approach infinite. Therefore,

In (

The structure of optical TDMA networks.

In Figure

For all interested simulative scenarios, the space between adjacent nodes is one slot length. Messages are similar in length. Every message can be contained in the payload of a slot.

The traffic distribution
among nodes affects the operation of traffic control in the MAC protocol. For
the benefit of easily performing traffic control, a basic traffic denoted by

In this paper, slot
frames are used to implement traffic control. The slot flow on optical fibers
is partitioned into frames. There are 1/

The simulation is applied to present the influence of the
adaptive maximum of a random variable on simulative precision. The waiting mean
of optical TDMA nodes manifests the precise degree caused by different adaptive
maximums. The simulative efficiency is dependent on the size of the adaptive
maximum. The larger the maximum of the random variable, the lower the efficiency
of the simulative system. On the other hand, the theoretic waiting mean
calculated by (

Two parameters
must be chosen before simulations. The parameters are the number of nodes

In simulations, two scenarios are interested. In order to certainly
manifest the effect of adaptive maximums, the

Relative parameters in two scenarios.

0.25 | 40 | 0.006411 | 300 | 46800 | 46807 |

3000 | 468000 | 468001 | |||

30000 | 4680000 | 4680001 | |||

50 | 0.005102 | 300 | 58800 | 58831 | |

3000 | 588000 | 588011 | |||

30000 | 5880000 | 5880023 |

For traffic control, the
number of slots in a frame, which is equal to 1/

In the following figures that show simulative results, the horizontal axis is the ordinal number of nodes. Because the ordinal number of nodes is discrete, all curves in figures consist of piecewise lines. The waiting mean of nodes on the vertical axis is expressed in slot times.

After simulations, (

Waiting means of 40-node optical TDMA networks.

Waiting means of 50-node optical TDMA networks.

Based on the theoretic
waiting mean,

300 | 3000 | 30000 | |
---|---|---|---|

40 | 27.40495 | 5.489945 | 2.63133 |

50 | 40.48827 | 8.349287 | 3.369718 |

The paper discusses tradeoff between simulative precision and efficiency. Based on the
statistical model of queuing delays, the difference between continuous and discrete probabilities is
used to manifest the effect of the maximum of random variables on the
statistical error. Then a simple method with precision parameter

In order to manifest the effect of the adaptive maximum on the
simulative precision, an optical TDMA network whose MAC protocol performs
traffic control is simulated. The average waiting time of some optical TDMA
node is in inverse proportion to the traffic of the node. The theoretic average
waiting time is exploited to calculate

In network simulations,
the adaptive maximum not only results in the acceptable degree of precision but
also suitably saves computing time. Based on the precision parameter