GNSS Availability Analysis in Taiwan - a Markov

Model ApproachHe-Sheng Wang

Department of Communications and Guidance Engineering

National Taiwan Ocean University, Keelung 202, TAIWAN

Phone: +886-912266254, FAX: +886-2-24633492

E-mail: hswang@mail.ntou.edu.tw

ABSTRACT

The main purpose of this paper is to present a Markov process approach to the analysis of GNSS (Global Navigation Satellite

System) availability in Taiwan. The proposed model is capable of calculating a variety of statistical measures of the required

services at locations specified by the user. The primary input data for the availability model is the MTBF (Mean Time Between

Failure) and failure rate of the GPS (Global Positioning System) satellites. Then each visible set of satellites is checked to see

if it satisfies the service accuracy criterion. Finally, in order to meet the integrity requirement, the number of the satellites and

their corresponding geometry are verified to see if they meet the integrity threshold. For other types of navigation systems con-

figurations (such as GBAS, GRAS, or SBAS) different service criteria may be added in the model. The analysis addresses GPS

constellation of 24 satellites, and augmentation with geostationay satellites or pseudolites. Availability estimates for GPS,

SBAS and GBAS are obtained by software simulation.

1. INTRODUCTION

The advent of the Global Positioning System (GPS) has the potential to revolutionize aircraft navigation technique. Although

GPS alone has the ability to provide the primary means of navigation services for en-route, terminal and non-precision ap -

proach phases of flight, augmentation is still needed to meet more stringent performance requirements on availability, accuracy

and integrity especially for the landing guidance during the precision approach phase. While the Satellite Based Augmentation

System (SBAS) is being implemented to provide seamless navigation guidance for all phases of flight down to Category I deci-

sion height of 200 feet, the purpose of the Ground Based Augmentation System (GBAS) is aimed at furnishing the aircraft with

high accurate landing guidance during the precision approach phase and all the way down to touchdown.

In the perspective of US Federal Aviation Administration (FAA), the objective of using GPS is to provide enhanced

service and to reduce infrastructure cost for aircraft navigation. To be used as a primary-mean navigation system, the Required

Navigation Performance (RNP) for integrity, availability, accuracy and continuity must be met. Primary means of navigation

refers to the capability of planning an operation around scheduled outages so that the system is available for a particular flight

and the operational continuity, availability and accuracy requirements are met. The four parameters for the RNP are defined in

the following.

1. Accuracy: The degree of conformance between the estimated or measured position and/or velocity of a platform at

a given time and its true position and/or velocity.

2. Integrity: The ability of a navigation system to provide timely warnings to user when the system should not be

used for navigation. Values stated are the probability that a system does not have integrity.

3. Continuity: The probability that the signal supports navigation accuracy and integrity requirements for the duration

of the intended operation (approach or landing), given that the capability was available at the beginning of the op-

eration, The continuity requirement is expressed as a loss of continuity per unit of time.

4. Availability: The probability that the navigation and fault detection functions are operational and that the signal ac-

curacy, integrity, and continuity of function requirements are met. This is typically express as a fraction of time.

The main purpose of this paper is to present a Markov process approach to the analysis of GNSS (Global Navigation

Satellite System) availability in Taiwan. The proposed model is capable of calculating a variety of statistical measures of the

required services at locations specified by the user. The primary input data for the availability model is the MTBF (Mean Time

Between Failure) and failure rate of the GPS (Global Positioning System) satellites. Then each visible set of satellites is

checked to see if it satisfies the service accuracy criterion. Finally, in order to meet the integrity requirement, the number of the

satellites and their corresponding geometry are verified to see if they meet the integrity threshold. For other types of navigation

systems configurations (such as GBAS, GRAS, or SBAS) different service criteria may be added in the model. The analysis

addresses GPS constellation of 24 satellites, and augmentation with geostationay satellites or pseudolites. Availability esti -

mates for GPS, SBAS and GBAS are obtained by software simulation.

2. MATHEMATICAL PRELIMINARIES

In this section, we briefly review some materials needed in the following development. Subjects to be exploited include the re -

liability theory and the descriptor systems theory.

The term “Availability” has its origin from the reliability theory. Therefore, to gain a further insight about how the avail -

ability of navigation systems can be defined, it would be beneficial to review some of the reliability theory.

When an item’s function is time dependent and it is needed throughout a time interval, its reliability is known as mission

reliability, expressed as

where R(t) is the reliability defined as the probability that the component will operate correct until time t, given that it was op-

erational at time 0; t is the specified mission time duration. More useful reliability parameters for continuously operating items

are the mean time between failures (MTBF) and the availability. The MTBF tells us how frequently, on the average, we can ex -

pect our item to experience an outage. The availability tells us the proportion of the time that we can expect our item to be op -

erating satisfactorily. Associated with these characteristics are the mean downtime (MDT), the average time that is takes to re-

turn to an operating state after an outage has occurred, and the outage rate, the complement of availability, or the portion of

time that we can expect our item to be down.

For reliability evaluation, we will be using several functions for describing the failure process. The key functions are

1. Probability density of failure, f(t)

2. Cumulative failure probability, Q(t)

3. Failure rate, (t)

The equation relating these functions are:

(1)

The failure density of the component, f(t), can also be expressed in terms of reliability as

(2)

The instantaneous failure rate function, , is given by

(3)

A well-know example of a probability density of failures is the exponential:

(4)

The choice of this density leads to a constant failure rate function . Of particular importance is the mean time to fail-

ure (MTTF), given by:

(5)

Often, the reliability of redundant system is expressed in terms of MTBF:

(6)

In view of our previous definition, the simplest model of the system availability can be derived as follows:

.

However, for a complex system such as GNSS navigation system, this model appears to be too rough.

A complex system is composed of a given quantity of elements, any of which may be up or down or in a standby mode at

any point in time. The following general statements about the complex system can be made:

1. At some point in time, say t=0, all elements are as designated to be. That is, elements designed to be normally operat-

ing are operating, those designed to be normally in a standby, those designed to be performing a monitoring function

are doing so, etc.

2. At any point in time, every element that is up (operating operable) will eventually be down.

3. At any point in time, every element that is down will some time be up (restored or replaced).

4. One-shot or cyclically operating elements can be said to be at t=0 in terms of a probability statement.

5. Every combination of elements up, down, standby comprises a different system state.

6. Upon a single failure or restoration event, the system passes form one state to another.

These statements describe a Markov process, the second type of Markov model. The states are discrete, defined by what

elements are up, down, standby. The time is continuous. Therefore the Markov model approach can be used for solving the

availability for complex systems.

Consider a general case consisting of a multitude of Markov success states, with transition among the states due to fail-

ure or restoration events. Let represent the rate from state j to state i. The transition rate diagram is shown in Figure 1. Let

us define all possible success states 1, 2,…, N and fail state F. Let be the probability of being in state i (i=1, 2,…, N, F)

at time t. Hence is the probability being in state i at time (t+t). Suppose that all states are mutually exclusive, then

we have

(7)

Define a time increment t that is so small that the probability of more than one event taking place during t is negligible. The

probability of being in a particular state i at time (t+t) is the probability that

1. The system is in state i at time t and it remains in state i during t.

2. The system is in some other state at the time t, and it goes from that state into state i during t.

Events 1 and 2 are mutually exclusive, so is the sum of the probability of event 1 plus the probability of event

2. Event 1 is the combination of two independent

Figure 1. A General Transition Rate Diagram

events, so its probability is the product of the probability of these two events. The probability that the system is in state i at

time t is ; the probability that the system remains in state i during the interval t is 1.0 minus the probability that it goes

from that i into another state during that interval—that is, 1.0 minus the rate at which it is drawn from state i times the interval

t. So that the probability of event 1 is . Similarly, the probability of event 2 is the probability that the

system is in some state other than i at start of the interval times the probability that it goes from that state into state i during the

interval t. Therefore the probability of event 2 is . Hence, the probability of being in state i at the time (t+t) is

(8)

(9)

Dividing both sides of equation (9) by t gives

(10)

Allow the notation to represent the sum of all transition rates out of state i:

(11)

Substituting equation (11) into equation (10) yields

(12)

Upon letting t approaches zero, Equation (12) then takes the following differential form:

(13)

From equation (7),

(14)

Hence, the availability at time t is the sum of the probabilities of being in each of the success states 1, 2, 3,…, N at time t.

In order to find the N values of from the N differential equation, we must have a boundary condition. Without loss

of generality, we can assume that

We can also solve with the system starting in any state i, not necessary state 1. A more general boundary condition

is

So the exact solution for the availability of a complex system with N success states over the time t is

(15)

where the values are the solutions to the set of N differential equations

(16)

In matrix format the availability of a complex system over time t can be obtained through the solution of the following state

space equation:

(17)

Note that, because the Pi’s constitute the complete probability space, it is necessary that the following condition is satisfied for

all t

(18)

Equations (17) and (18) constitute a descriptor system. Before introducing the elementary facts about descriptor systems,

we first consider a degenerate case of Equations (17)-(18) in the following. Equation (17) describes the instantaneous availabil-

ity. There is another type of the system availability, namely the steady-state availability. To obtain the steady-state availability,

we simply take the limit of equation (17) as t approaches infinite:

(19)

(20)

where the values of Pi are the solutions to the set of independent, linear equations:

(21)

To avoid a trivial solution, for the most trivial equation (usually the last one) in the set we can substitute

(22)

Equation (22) is valid because all states are mutually exclusive and exhaustive. Equations (21) and (22) can be put in a more

compact form as follows:

(23)

The derivation of Equations (18) through (23) in the preceding paragraph demonstrates that steady-state availability is

(24)

Let us now get back to the solution of the equations (17)-(18) which can be put in the following descriptor form:

(25)

where

,

and u(t) is the unit step function. Equation (25) is different from the ordinary differential equations in that the matrix E is sin-

gular, in general. We have the following theorem concerning the solvability of the descriptor systems.

Theorem 1. The descriptor system (25) has a unique solution for a given input function u(t) and any initial condition of the

form EP(t0) if and only if the matrix pair {E, F} is regular, i.e., there exists a scalar 0 such that 0E-F is not identically zero.

The last equation in (25) describes an algebraic constraint among the state variables and the input function, which is the

key to the analysis of GNSS availability. Specifically, if we want to develop different Markov models for various GNSS aug-

mentation systems (such as SBAS, GBAS, GRAS, etc.), we simply add a set of appropriate algebraic constraints to the descrip-

tor systems, and construct a set of input functions so that solution trajectories of Equation (25) can be arbitrarily assigned. This

concept exactly coincides with the notion of the controllability of the descriptor systems.

Theorem 2. The descriptor system (25) is completely controllable if and only if rank[sE-F B]=N for all complex scalar num-

bers s.

3. MARKOV MODEL APPROACH TO SOLVING COMPLEX SYSTEM AVAILABILITY

The general information flow and computation diagram is depicted in Figure 2. This kind of computational diagram was first

proposed by Poor [4]. Before we can use the Markov model to compute the system availability, we need to derive an effective

system MTBF in order to reflect the fact that, for different satellites, MTBF should not be the same, in general.

Recall our earlier results:

(26)

Let t be the time required for the system to get from state 1 at time t=0 to the absorbing state F. The mean of the variable t,

which is the MTBF,

(27)

where is the moment- generating function. Notice that f(t) is the failure density function for

the system. Because t must be positive, the moment-generating function becomes

(28)

Recall the definition of the Laplace function of a time function t

(29)

which is exactly the moment-generating function with = -s. Hence,

(30)

where the failure density function is

(31)

and from Equation (24) we saw that

(32)

Hence,

(33)

from which follows

Figure 2. Information Flow Diagram for GNSS Availability Analysis

(34)

From Laplace transform theory,

(35)

By the boundary condition：

(36)

Hence,

(37)

Form which we conclude

(38)

To serve as a illustrated example, we consider here a basic scenario of the availability model. We consider a 24 GPS

satellite constellation with the following parameters:

Almanac Data: Friday, September 10, 2004 10:53 AM GMT Song-Shan Airport Location: N(25° 04’ 17”), E(121° 32’ 36”) Week Number: GPS 264+1024 Week GPS Time (into week): 386460 sec Mask Angle: 5 degree System Mission Time: 500 Hour Satellite Failure Rate: 1/3394 = 294.64*10-6 per hour Satellite Restoration Rate: 1/20 = 0.05 per hour

The system reliability diagram is shown in Figure 3, and the corresponding success states and failure state is shown in Table 1.

The computed availability for a single time epoch is:

Availability (GPS) = 0.9984632074.

4. SIMULATION RESULTS

In the section, simulation results for SBAS and GBAS systems availability are given. The main parameters are the same as in

the previous section. Basic assumptions are similar to GPS Risk Assessment Study reported by Johns Hopkins University [1].

For SBASE case, a 24 satellites constellation plus a geostationary satellite that has full coverage of Taipei Information Region

(FIR) is assumed to-

Figure 3. Block Diagram for GPS Availability Analysis

gether with a possible deployment of the ionospheric grid points. At the beginning of the simulation, the in-view satellites and

their ECEF coordinates are given in Figure 4 and Table 2, respectively.

,

5 of 8 require

,

,

,

Table 1. Success State and Failure State Definition

The geometric dilution of precision (GDOP) of the in-view satellites is incorporated into the descriptor system (25) as a

weighting matrix to partly reflect the accuracy requirement for GNSS navigation system. For a given set of in-view satellites,

the weighting matrix can be computed as follows. Assume the visible satellites are given as in Table 2. We first compute a vec -

tor of partial GDOPs. The results are shown in Table 3. Then a weighting matrix can be derived as follows:

Figure 4. Visible Satellites at the Initial Epoch

X Y Z

PRN 3 -1.8142e+007 -1.5163e+006 1.9313e+007

PRN 8 5.5144e+006 1.641e+007 1.985e+007

PRN 11 -1.3535e+007 2.2529e+007 3.1785e+006

PRN 13 5.6225e+006 2.4257e+007 -9.391e+006

PRN 16 -2.6473e+007 -1.8316e+006 -1.3719e+006

PRN 19 -1.1716e+007 1.0497e+007 2.1513e+007

PRN 27 -8.2794+004 2.2177e+007 1.4764e+007

PRN 31 1.4707e+007 2.1864e+007 -1.7271e+006

Table 2. ECEF Coordinates of the In-view Satellites

-------------------------------------------------------------Success Fail Items Items Items under State State UP Down Restoration---------------------------------------------------------------------------

- 01 08 00 0 02 07 01 0 03 06 02 0 04 05 03 1**************************************************

* 05 04 04 1 06 03 05 2 07 02 06 2 08 01 07 2 09 00 08 2---------------------------------------------------------------------------

--

WGDOP =

diag[GDOP1, GDOP2, GDOP3, GDOP4, GDOP5, GDOP6, GDOP7, GDOP8] = diag[1.9462, 1.8911, 2.7357, 1.8862,

2.1463, 1.8364, 1.8732, 2.0776].

SBAS model analysis area is shown in Figure 5. The area enclosed by the red line indicates the assumed SBAS coverage

volume. Simulation result for SBAS system availability at the Song Shan airport is shown in Figure 6.

For GBAS system availability, the main parameters are the same as in the previous section. GPS/GBAS analysis baseline

assumed a 30 GPS satellites constellation with 2 airport

PRN 8、11、13、16、19、27、31 GDOP1 1.9462

PRN 3、11、13、16、19、27、31 GDOP2 1.8911

PRN 3、8、13、16、19、27、31 GDOP3 2.7357

PRN 3、8、11、16、19、27、31 GDOP4 1.8862

PRN 3、8、11、13、19、27、31 GDOP5 2.1463

PRN 3、8、11、13、16、27、31 GDOP6 1.8364

PRN 3、8、11、13、16、19、31 GDOP7 1.8732

PRN 3、8、11、13、16、19、27 GDOP8 2.0776

Table 3. Partial GDOPs: Each of GDOPi is Computed by Excluding One Satellite from the Observed Set

Figure 5. SBAS Analysis Area

pseudolites. GPS/GBAS accuracy models were based on the specifications given in RTCA/DO-245. For GBAS analysis, GPS

signal integrity is first investigated based on a 24 hours observation data collected in Song Shan airport. Statistical results are

given in terms of the protection levels which are shown in Figure 8 in the next page.

Figure 6. SBAS Availability vs. GPS Time

Availability analysis for GBAS model is given in Figure 7. A world-wide GBAS availability simulation is also per-

formed. For each location, number of the satellites in-view and the coordinates of the satellites are computed by using the soft-

ware package “Satellite Navigation TOOLBOX for Matlab,” by GPSoft LLC. The result is shown in Figure 9 in the next page

and also summarized in Table 4.

5. CONCLUSIONS

In this paper, we have proposed a Markov process approach to the analysis of GNSS availability. Both SBAS and GBAS con -

stellations are considered. Based on the assumptions similar to [1], the results are also compatible with it.

Figure 7. GBAS Availability vs. GPS Time

No. City Latitude Longitude Availability

1 London 52°N 0°W 0.99999

2 Liberia 7°N 10°W 0.9999

3 Iceland 65°N 22°W 0.9999

4 Buenos Aires 30°S 58°W 0.99999

5 Ecuador 3°S 80°W 0.9999

6 Winnipeg 50°N 95°W 0.99999

7 Los Angeles 34°N 118°W 0.99999

8 North Alaska 70°N 150°W 0.9999

9 Honolulu 22°N 158°W 0.9999

10 Ross Sea 75°S 180°W 0.9999

11 Tokyo 36°N 140°E 0.9999

12 Perth 32°S 115°E 0.99999

13 Singapore 2°N 75°E 0.9999

14 Indian Ocean 45°S 60°E 0.9999

15 Aral Sea 45°N 50°E 0.99999

16 Cape Town 35°S 18°E 0.9999

Table 4. World-Wide GBAS Availability Analysis

REFERENCES

[1] T.M. Corrigan et al., GPS Risk Assessment Study Final Report, The Johns Hopkins University Applied Physics Laboratory,

January 1999.

[2] Per Enge, “Local Area Augmentation of GPS for the Precision Approach of Aircraft,” Proceedings of the IEEE, Vol. 87,

No. 1, pp. 111-132, 1999.

[3] P. Kales, Reliability for Technology, Engineering, and Measurement, Prentice-Hall, Upper Saddle River.

[4] W. A. Poor, “Description of a GNSS Availability Model and Its Use in Developing Requirements,” IEEE Trans. on Aero-

space and Electronic Systems, Vol. 31, No. 1, pp. 436-446, 1995.

[5] RTCA/SC-159, Minimum Aviation System Performance Standards for the Local Area Augmentation System (LAAS),

RTCA DO-245, 1998.

[6] “Satellite Navigation TOOLBOX 2.0 for Matlab”, GPSoft LLC, 1999.

[7] Yeou-Jyh Tsai, Wide Area Differential Operation of The Global Positioning System: Ephemeris and Clock Algorithms, Ph.

D. Dissertation, Dept. of Aeronautics and Astronautics, Stanford University, 1999.

[8] H. S. Wang, C. F. Yung, and F. R. Chang, H∞ Control for Nonlinear Descriptor Systems, Lecture Notes in Control and In-

formation Science, Springer-Verlag, New York, 2006.

Figure 7. Song Shan Airport GPS Signal Availability

Figure 8. GBAS Availability vs. Location