Accuracy is the ability of the system to predict future events. A higher degree of accu- racy will result in more “cache hits” of the predicted state cache information. Smaller tolerances should result in greater system accuracy, but this comes at the cost of a re- duction in speedup.
Assume for simplicity that the effects of non-causality are negligible for the analy- sis in this section. The effects of causality are discussed in more detail in Section 4.7.2. An Logical Process may deviate from the real object it represents because either the Logical Process does not accurately represent the actual entity or because events out-
0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 VNC Speedup
Proportion of Out-of-Tolerance Messages (Y) VNC Speedup Analysis
Speedup
side the scope of the predictive system may effect the entities being managed. Ignore events outside the scope of the predictive system for this analysis and consider only the deterministic error from inaccurate prediction of the driving process. The error is defined as the difference between an actual message value at the current time (
v
t) and a message value which had been predicted earlier (v
tp). Thus the Message Error isME
= v
t ,v
tp. Virtual message values generated from a driving process may containsome error. It is assumed that the error in any output message generated by a process is a function of any error in the input message and the amount of time it takes to process the message. A larger processing time increases the chances that external events may have changed before the processing has completed.
Two functions of total ACcumulated message value error (
AC(
)
) in a predicted re-sult are described by Equations 4.40 and 4.41 and are illustrated in Figure 4.22.
ME
lp0
is the amount of error in the value of the virtual message injected into the predic- tive system by the driving process (
lp
0). The error introduced into the value of theoutput message produced by the computation of each Logical Process is represented by the Computation Error function
CE
lpn(ME
lpn,1;
t
lpn)
. The real time taken for then
th Logical Process to generate a message ist
lpn. The error accumulates in the StateQueue (SQ) at each node by the amount
CE
lpn(ME
lpn,1;
t
lpn)
which is a function of theerror contained in the input message from the predecessor Logical Process and the time to process that message. Figure 4.22 shows a driving process (
DP
) generating a virtual message which contains prediction error (ME
lp0). The virtual message with predictionerror (
ME
lp0) is processed by nodeLP
1int
lp1 time units resulting in an output message
with error,
ME
lp1= CE
lp0(ME
lp0 ;t
lp1
)
.Proposition 4 The accumulated error in a message value is Equation 4.40 and Equa-
tion 4.41.
LP1 LP2 ME lp1 ME lp0 lp1(MElp1,t lp2) CE (ME lp0, t lp1) CElp0 λv Driving Process Virtual Message LP 0
(virtual message generation rate)
Figure 4.22: Accumulated Message Value Error.
AC
n(n) =
N X i=1CE
lpi(ME
lpi,1 ;t
lpi)
(4.40)AC
t(
) = lim
P tlpi! lpn X i=1CE
lpi(ME
lpi,1 ;t
lpi)
(4.41)where
CE
lpi is the computational error added to a virtual output message value,ME
lpi is the virtual message input error, andt
lpi is the real time taken to process avirtual message.
As shown in Proposition 4,
AC
n(n)
is the total accumulated error in the virtual message output by then
th Logical Process from the driving process.AC
t(
)
is theaccumulated error in real time units from the generation of the initial virtual mes-
sage from the driving process. Equation 4.41 is
lim
P tlpi!Plpn
i=1
AC
n(n)
, where
lpn
is the number of Logical Process computations in time . In other words,AC
t(
)
isFor example, if a prediction result is generated in the third Logical Process from the driving process, then the total accumulated error in the result is
AC
n(3)
. If 10 repre- sents the number of time units after the initial message was generated from the driving process thenAC
t(10)
would be the amount of total accumulated error in the result. A cache hit occurs whenjAC
t(
)
j6, whereis the tolerance associated with the lastLogical Process required to generate the final result. Equations 4.40 and 4.41 provide a means of representing the amount of error in a Virtual Network Configuration gen- erated result. Once an event has been predicted and results pre-computed and cached, it would be useful to know what the probability is that the result has been accurately calculated, especially if any results are committed before a real message arrives. The out-of-tolerance check and rollback does not occur until a real message arrives. If a resource, such as a Virtual Circuit, is established ahead of time based on the predicted result, then this section has defined
= P[
jAC
t()
j>]
where is the toleranceassociated with the last Logical Process required to generate the final result.
4.4.1
Rapidly Deployable Radio Network Error Accumulation
In the Rapidly Deployable Radio Network Network Control Protocol implementation, no network resources are committed, except for pre-loading the beam table, until the real message for an event is received. Assume the beamform table has an estimated download time of five milliseconds and a one degree tolerance for error in position. In the implementation used for the experimental validation, there is only one level of error propagation, and thus no error accumulation. Thus, is shown in Equation 4.42 as
illustrated in Figure 4.23 where
d
is the distance between the nodes, 2is the variance in location prediction accuracy, and
is the tolerance. Anglee
in Figure 4.23 is the variance of error in the beam steering angle due to the variance in the Cartesian co-
Pr
tan
,1d
> (4.42)ordinates provided by the prediction process.
RS-232 PCI PCI Pckt Radio GPS Rec Radio Antenna OC-3c EN RS-232 PCI Pckt Radio GPS Rec Radio Antenna RN σ2 d e
Figure 4.23: Beam Steering Variance.