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We consider a real-world problem of military intelligence unit equipped with multiple identical unmanned aerial vehicles (UAV) responsible for several regions (with requests of real-time jobs arriving from independent sources). We suppose that there are no ample maintenance facilities, allowing simultaneous treatment of all vehicles if necessary. Under certain assumptions, these real-time systems can be treated using a queueing theory methodology and/or as Markov chains. We show how to compute steady-state probabilities of these systems, their performance effectiveness, and various performance parameters (for exponentially distributed service and maintenance times of UAVs, as well as tasks duration and their arrival pattern).

From the earliest days of warfare, military commanders have wanted to know what lies over the hill. Today, the battlefield usually holds no secrets from sophisticated flying platforms. Modern airborne reconnaissance structures rely on a combination of satellites, aircraft and unmanned aerial vehicles (UAV). According to recent concept, a real-time data collected by different systems would be further integrated and redistributed in framework of Network-Centric Operations system. It would assimilate the data, recognize and control events, create a mosaic of what is happening at any time and provide a real-time decision support [

To perform these functions properly, the best of modern technologies and methodologies must be used. High on the list of favored options are unmanned aerial vehicles (UAV’s). It is difficult to overestimate their role in real-time intelligence gathering, round-the-clock surveillance and day/night reconnaissance operations. These aircrafts are indispensable in monitoring restricted, hard-to-reach and dangerous locations.

During last two and half decades, various models concerning UAV’s have been presented in scientific literature.

In [

In [

The use of military systems involving UAVs relies on the principle of availability, i.e. their ability to process the maximal portion of real-time tasks. Traditional definitions of availability are not compatible for complex (e.g. multichannel) systems where changes of performance levels need not be identified with system failure. In [

In this work, we study the problem of multiple UAVs operating in general regime with limited maintenance facilities (extension of [

We consider a military intelligence unit equipped with N identical UAVs responsible for r non-overlapping homogeneous reconnaissance regions required to be under surveillance. The military command sends orders/tasks to patrol the region in real time (e.g. 9:00 - 10:00) without advance notice. If all UAV are not available until 9:30, then only the second half of the order/task gets filled. To observe one region at any moment, only one UAV is needed, thus no additional orders are sent to the region, which is already under observation. Therefore one UAV at most is used (with others being in maintenance or on stand-by or providing the service to another region) to execute the order concerning this region at any moment. Thus, the total number of orders in this military unit cannot exceed the number of regions, i.e. r. Execution of an order, which has found an available UAV starts immediately upon its arrival and continues while where are available UAVs in the system. Different parts (time intervals, e.g. 9:00 - 9:15 and 9:20 - 9:50) of the same order can be executed by different UAVs. Any part of the order that is not executed immediately (e.g. 9:15 - 9:20) in real time is lost. Queues of orders or their parts do not exist in this system.

An UAV flying over any region is operative for a period of time

In this section we suppose that: UAV’s operation time (time to failure TTF), its maintenance time, orders inter-arrival and durations times are exponentially distributed. This enables as to treat the model under consideration as a Markov chain.

We define the state of the system (m, n), with

To be more specific we assume that:

UAV’s operation times are i.i.r.d.v.

There are no additional order arrivals when there are already r orders in the system.

Now we can calculate the numbers of UAVs, regions, orders and maintenance facilities in different positions in terms of m and n, namely:

Number of UAVs out of order is N − n;

Number of UAVs in maintenance (broken) is min(K, N − n) (busy facilities);

Number of UAVs waiting for maintenance (broken) is max(0, N − n − K);

Number of idle maintenance facilities is max(0, K − N + n);

Number of operating UAVs (executing one of orders) is k = min(m, n);

Number of regions under surveillance (executed orders) is also k = min(m, n);

Number of UAVs on stand-by (fixed) is n − k = max(0, n − m);

Number of regions with no order (empty) is r − m;

Number of non-executed orders waiting for service (task’s time is expiring) is m − k = max(0, m − n).

A corresponding set of simultaneous linear equations for steady state probabilities is as follows:

for the corner state m = n = 0;

for the corner state m = 0, n = N;

for the corner state m = r, n = 0;

for the corner state m = r, n = N;

for “interior” states of diagram 0 < n < N, 0 < m < r;

for “interior” states on the upper border of diagram m = 0, 0 < n < N;

for “interior” states on the left border of diagram n = 0, 0 < m < r;

for “interior” states on the lower border of diagram m = r, 0 < n < N

for “interior” states on the right border of diagram n = N, 0 < m < r; and finally

Thus we have proved the following:

Theorem 1: The steady state probabilities of system under consideration, with N UAVs, r regions, K < N maintenance facilities and independent exponentially distributed TTF, maintenance, inter-arrival (arrival rate proportional to the number of regions with no order) and order duration times is the unique solution of the system of linear Equations (1)-(10).

The system of linear Equations (1)-(10) can be easily solved by standard procedures.

Note: In the system under consideration the backlog of orders/tasks (total time of all orders at any instant) is not influenced by UAVs/servers and maintenance teams since order’s time is expiring anyway (either being processed or lost). Therefore, some of the probabilities (namely

It can be easily transformed to

Thus the number of orders in the system is distributed binomially.

Performance Effectiveness Index (PEI) characterizes a system ability to perform its main functions even with partial capacity, and is defined [

For systems under consideration the following definitions were suggested [

Definition 1: The current effectiveness of the system at moment u:

W (u) = (number of operating UAVs at moment u)/(number of orders at moment u).

Definition 2: Performance Effectiveness Index (PEI) of the system is the expected value of the current effectiveness, namely:

It was shown [

Ignoring the case, when there are no orders in the system, i.e. m = 0.

Assuming automatically the maximal effectiveness 1 in the case, when there are no orders in the system (m = 0), even if there are no fixed UAVs in the system. And

by-passing this complications (no order in the system, m = 0)

It is important to remember, that the probabilities

Next we shall show how to calculate some useful performance characteristics of the system under consideration.

Each UAV can be in one of four positions at any moment:

i) fixed and operating;

ii) fixed on stand-by;

iii) in maintenance;

iv) waiting for maintenance (shortage of facilities).

Each region can be in one of three positions at any moment:

i) with processed order inside;

ii) with non-processed order inside;

iii) empty (m = 0).

Each maintenance facility can be either idle or busy at any moment.

Each order/task can be either processed or non-processed at any moment.

Now we can obtain corresponding average values (see Section 3 above):

for average number of fixed UAVs;

for average number of operating UAVs (also an average number of processed orders);

for average number of fixed UAVs on stand-by;

for average number of broken UAVs;

for average number of UAVs in maintenance (also an average number of busy maintenance facilities);

for average number of idle maintenance facilities;

for average number of UAVs waiting for maintenance;

number of regions needed to be under surveillance);

for average number of non-processed orders;

for average number of empty channels.

Finally, the performance measure can be introduced via cost function. Let C be the cost of one processed order during the time unit, D be the cost of one non-processed order (lost part) during the time unit, G be the cost of one UAV being on stand-by during the time unit, H be the cost of one UAV being in maintenance during the time unit, F be the cost of one UAV waiting for maintenance during the time unit and Q be the cost of one idle maintenance facility during the time unit.

Then the total expected cost per time unit of system operation is given by the following formula

This formula allows to make a good choice of numbers of UAVs and maintenance facilities needed for the proper operation of the system.

In this Section we present some numerical results for N = 9, r = 5 and different sets of values.

It can be easily seen from the numerical results that all three PEI’s:

i) increase, when

ii) decrease, when

iii) increase, when

iv) decrease, when

\m l\ | 1 K = 9 | 2 K = 9 | 1 K = 8 | 2 K = 8 |
---|---|---|---|---|

1 | 0.9900 0.9903 0.9850 | 0.9604 0.9616 0.9286 | 0.9851 0.9856 0.9775 | 0.9407 0.9424 0.8928 |

2 | 0.9992 0.9992 0.9986 | 0.9877 0.9881 0.9804 | 0.9988 0.9987 0.9980 | 0.9816 0.9822 0.9701 |

\m l\ | 1 K = 9 | 2 K = 9 | 1 K = 8 | 2 K = 8 |
---|---|---|---|---|

1 | 0.9987 0.9989 0.9976 | 0.9786 0.9814 0.9655 | 0.9982 0.9984 0.9964 | 0.9679 0.9723 0.9481 |

2 | 0.9999 0.9999 0.9998 | 0.9979 0.9982 0.9961 | 0.9999 0.9999 0.9997 | 0.9969 0.9973 0.9941 |

\m l\ | 1 K = 9 | 2 K = 9 | 1 K = 8 | 2 K = 8 |
---|---|---|---|---|

1 | 0.9678 0.9679 0.9590 | 0.8776 0.8781 0.8521 | 0.9515 0.9517 0.9385 | 0.8164 0.8172 0.7781 |

2 | 0.9970 0.9970 0.9959 | 0.9650 0.9651 0.9546 | 0.9955 0.9955 0.9939 | 0.9475 0.9476 0.9271 |

In this paper, we presented a real world problem concerning multiple UAVs. Number of orders in this system is binomially distributed and does not depend on UAVs and maintenance facilities, since order/task time is expiring anyway (either being processed or lost). We presented this system as a Markov chain and provided a set of linear equations for steady-state probabilities, as well as performance effectiveness index, average cost function and other performance characteristics.

Joseph Kreimer, (2016) Performance Evaluation of Multiple Unmanned Aerial Vehicles Operating in General Regime with Shortage of Maintenance Facilities. Journal of Computer and Communications,04,70-78. doi: 10.4236/jcc.2016.410008