It is anticipated that future skies over urban areas will be busy with drones flying back and forth delivering packages. Taking New York City as an extreme example, it is estimated that by 2026, 2600 delivery drones could simultaneously populate the city's airspace. The drone–drone collision rate of “dumb” drones can be calculated by treating them as a gas of large, randomly moving, spherical molecules, using the kinetic theory of gases. Collisions can be avoided by making each drone “smart,” i.e., by giving each a “sense and avoid” capability for detecting and avoiding a potential collision. For smart drones over New York City, the rate of potential collisions, or encounter rate, extends over a surprisingly large range: from 1 to 170,000 encounters/day, depending on input assumptions. This places stringent constraints on the probability that a smart drone encounter will result in a collision, constraints that must be met by the drone operator. Policy implications are discussed.

## I. INTRODUCTION

Drones can deliver packages faster and at lower cost than street-level vehicles, especially for one vehicle trip per package.^{1,2} But drone delivery faces numerous challenges: concerns about safety,^{3–5} noise,^{6} a possibly unwelcoming public,^{7} and a several year pathway to achieving all the certifications required for full-scale operation.^{8} Given the speed and cost advantages of drone delivery, but still cognizant of the challenges, a number of companies have been developing and testing drone delivery systems. These include: Amazon Prime Air, Wing (a subsidiary of Alphabet), UPS Flight Forward (with Wingcopter), DHL, FedEx, Zipline, Flirtey, Matternet, and Uber Eats.^{3,9–12} Gartner, a consulting firm, predicts that in 2026 more than one million drones will be engaged in retail deliveries worldwide.^{1} For that same year, FedEx expects the U.S. parcel market alone to exceed 100 × 10^{6} packages per day, including drone deliveries as well as vehicle deliveries.^{13} Ninety percent of these could be deliveries by drone.^{3,9}

This envisioned future of drone delivery will only be realized if delivery by drone is safe. Collisions with buildings, towers, wires, flying taxis, passenger drones, helicopters, winged aircraft, and other delivery drones must be avoided. We will use the kinetic theory of gases to calculate rate of collisions between “dumb” drones that have no collision-avoidance capability, modeling the drones as large molecules. Such a capability is needed to reduce the number of collisions to zero or near-zero. We distinguish two strategies for achieving this:

Equipping each drone with a “sense and avoid” capability to detect an imminent collision and then swerve to avoid it. Amazon's Prime Air delivery drone

^{4,9}(see Fig. 1) utilizes this strategy.Creating an Unmanned Aircraft System Traffic Management (UTM) system to ensure a safe separation between drones, using intelligence resident outside the drones themselves. A UTM is currently under development by the FAA and NASA, working in collaboration with other federal agencies and industry, but will only become well-defined over the next few years.

^{14,15}Because of the lack of specificity, we will not consider this strategy further in this paper.

The measure of risk is the rate of collisions (collisions per day) CPD. The collision calculations in this paper are based on strategy 1, for which we have

where EPD denotes encounters per day and *p* is the probability that an encounter becomes a collision. An encounter occurs when two drones approach each other and will collide unless one or both swerve to avoid the collision. In the Prime Air drone case, Amazon will determine *p* by repeatedly flying two drones on intersecting paths. (Amazon has previously tested its drones via simulations, not by actual physical tests as far as the author has been able to determine.) We estimate EPD for plausible levels of drone activity over a large urban area: New York City. Municipal authorities will be responsible for specifying CPD_{acc}, the maximum acceptable level of CPD (presumably zero or near zero). For an acceptable drone delivery service, *p* ≤ *p*_{max} = CPD_{acc}/EPD, a constraint the drone operators must be able to meet. The goals of this paper are: (i) to estimate EPD by treating the cloud of drones as a gas of randomly moving spherical “molecules”; (ii) to obtain values of *p*_{max} from the range of estimated EPD values; (iii) to estimate the extent of drone crowding in the city's skies, in particular, the average inter-drone separation; and (iv) to discuss policy implications of (i)–(iii). There are two limitations to the kinetic theory approach. (a) If drones in a portion of the cloud all have the same speed, the mean-free-path formula of kinetic theory must be modified. (b) Assuming drones to be spherical, each with a radius equal to half its longest dimension, overestimates their collision rate; we will compensate by considering a range of drone radii.

## II. ANALYSIS

If there are many drone flights over a city, but only one drone at a time is in the air, there will be no collisions. If many are in the air at one time, but all travel on radial paths from a central point, the outgoing drones at one altitude and the incoming at a lower, there will be no collisions, assuming care is taken to ensure that at the central point rising and descending drones do not conflict. However, if Amazon drones are launched from and return to one central point and Wing drones from a different central point, the Amazon and Wing drones, flying at the same altitude, could collide. We assume many such central points, roughly distributed uniformly over the area of New York City.

We refer to drones that have no detect and swerve capability as “dumb” drones, in contrast to “smart,” strategy 1 drones that do have that capability. Real-world delivery drones will all be smart. But will they be sufficiently smart so that *p* is small enough to satisfy $EPD\xd7p\u2264\u2009CPDacc$? We assume that the encounter rate EPD is the same for smart and dumb drones, and that, for dumb drones, each encounter results in a collision, so $CPDdmb=EPD.$

For our dumb drones, we assume a radius *a *=* *1 m, the approximate half-length of Amazon's Prime Air drone.^{5,6} A molecule traveling at speed *v* will sweep out a volume $\pi 2a2vt$ in time *t*, colliding with $\pi 2a2vtN$ molecules in the process, where 2*a* is the effective molecular radius and *N* is the number of molecules per unit volume. The effective radius is needed because molecular centers must be more than two radii apart to avoid collisions. Dividing *vt* by the number of collisions in time *t* yields the average distance per collision or the mean free path λ^{16}

Equation (2) assumes all molecules in the swept-out volume are stationary, except for the one moving at speed *v.* If the molecules are not stationary, but have a Maxwellian distribution of velocities, a more accurate expression for the mean free path is found^{16}

For molecules that all have the same speed *v* (like drones, more or less), but move in random directions, a drone-appropriate expression for the mean free path is

Equation (4) was derived by Clausius^{16} and is used in the following calculations. Since λ_{Cl} for moving molecules is only 25% less than λ for stationary molecules, the fact that drones in the drone cloud do not have a completely random velocity distribution does not much matter. Their spatial density *N* is more important.

If *V* is the volume of the drone cloud and *P* is the total number of drones in the cloud, then

In the USA, commercial drones cannot be flown above an altitude of 400 ft (122 m) without special permission from the FAA.^{17} We assume a vertical height of 100 m (328 feet) for the permissible drone fly zone: 22 m–122 m. This is more generous than allowed by a bill introduced in the US Senate on October 16, 2019. That bill would give a property owner control of the airspace up to 200 ft (61 m) over their property, resulting in a commercial drone fly zone thinner than the 100 m vertical range assumed here and consequently a higher drone density. At 100 m thickness, the commercial drone fly zone is still a very thin slice of air (punctured by tall buildings) sandwiched between the domain of commercial aircraft, light aircraft, air taxis, and helicopters, and the zone below. New York City^{18} has an area of 785 km^{2}. With a thickness of 100 m, the drone cloud over the city has volume *V *=* *78.5 km^{3}.

Assuming a steady state

where *μ* is the average number of drones/hour launched into the drone cloud and *W _{s}* is the average round trip flight time in hours. For example, if 12 drones are launched per hour and the average time of flight is ½ hour, from when a drone is launched until its return, an average of six drones will populate the sky at any time.

Of the 1.5 × 10^{6} packages delivered per day in NYC in 2019,^{19} let us assume that 10%, or 150,000, are delivered by drones. Ten percent is a conservative estimate, according to Jeff Wilke, Amazon's CEO of worldwide consumer services. At Amazon's June 5, 2019 conference in Las Vegas, Wilke unveiled the company's latest Prime Air drone, which can fly up to 15 miles (24 km) round trip to deliver packages of less than five pounds within 30 min.^{3,5} Wilke stated: “And while 5 pounds may not sound like a lot, it represents between 75 and 90 percent of the packages that Amazon delivers to its customers today.” If 150,000 drone deliveries are made per day, and there are no deliveries between midnight and 6 a.m., then *μ* = 8333 drone launches/hour. Assuming that 10 min are devoted to preparing the package and loading it into the drone, then *W _{s}* = 20 min. With

*a*=

*1 m,*

A drone flying out for 10 min and back for 10 min will, at a speed of 30 mph (48 km/h), travel 10 mi (16 km) round trip. The probability of a collision in one flight, with this average round-trip distance *d* = 16 km, is *d*/λ_{Cl} = 0.009.

Note that 16 km is 2/3 of the maximum 24 km round trip distance claimed for the Prime Air drone and is the average delivery round trip distance assuming destinations have a uniform spatial distribution over a circle of radius 12 km. To see this, imagine a drone base station at the center of its circular delivery area, with radius *R* = the maximum one-way drone range, and destinations distributed uniformly throughout the delivery area. The average base-to-delivery distance is then $r\u0302=1/\pi R2\u222b0Rr2\pi rdr=(2/3)R$.

With 150 000 drone deliveries per day over New York City, and *d *=* *16 km per delivery, the total distance traveled by the drone fleet per day will be $2.4\xd7106$ km, or 1400 *λ*_{Cl}, implying 1400 dumb drone collisions/day. However, this counts the collision of the *i*th drone with the *j*th, and the *j*th with the *i*th, as two distinct collisions. Dividing by two to correct for this double counting yields, for dumb drones, a rough estimate of CPD_{dmb} = 700 collisions/day. (When the author first arrived at this estimate, he was surprised that it was so large, having anticipated very few encounters, as there is much empty space above New York City.) Clearly, smart drones must be much better at avoiding collisions. If we arbitrary take CPD_{acc} = 10 collisions/year = 2.74 × 10^{−2} collisions/day, then to meet the condition $EPD\xd7p\u2264CPDacc$, *p *≤ 0.39 × 10^{−4}. The collision-avoidance capability of smart drones must reduce the collision rate by a factor of at least 1/p = 2.6 × 10^{4}. Meeting this condition is the responsibility of drone manufacturers and will be a non-trivial challenge.

We can express these results mathematically by combining Eqs. (4)–(6). Defining LPD as launches per day, we have

and

where *s* is the drone speed, the substitution *W _{s}* =

*d*/

*s*has been employed, and the factor of 2 in the denominator of Eq. (9) eliminates double counting. From Eq. (9), EPD can be reduced by decreasing the round-trip flight distance

*d,*by reducing the drone size

*a,*or by increasing the speed

*s.*Seven hundred encounters/day can be taken as a practical upper bound. We obtain a lower bound for EPD by taking

*d*=

*4 km (much less than the 16 km average round-trip range of Amazon's Prime Air drone),*

*a*=

*0.5 m (the wingspan of Wing's drone is 1 m), and*

*s*=

*113 km/h (the top speed of Wing's drone). Using these values reduces EPD by a factor of 150, yielding a lower bound value of 4.7 encounters/day, or*

*p*≤ 0.583 × 10

^{−2}, a collision probability range much less challenging than the value found earlier.

As noted above, far more than 10%, up to 90%, of deliveries could be made by drone.^{3,4} From Eq. (9), EPD varies as *μ*^{2}, because as *μ* increases, the cumulative distance traveled increases while the mean free path decreases. If 50% of packages are drone-delivered, EPD will be 25 times greater than for 10%. Table I shows the results of our analysis if 10%, 50%, and 90% of all deliveries are made by drone. For LPD = 150,000 over New York City, the drone density *N *=* *35.4 drones/km^{3}. The procedure for calculating the average separation between drones is presented in the Appendix.

According to Ref. 13, the U.S. parcel market is expected to double in size, to more than 100 × 10^{6} packages per day by 2026. New York City's population is 2.6% of the U.S. population, so it can be expected to have roughly 2.6 × 10^{6} deliveries/day by 2026. Assuming again 10% of these NYC deliveries are by drone, there will be 2100 drone encounters/day in 2026, 53,500 at 50% drone delivery, and 173,000 at 90% drone delivery (upper bounds). Table II shows the same variables as Table I, but for 2.6 × 10^{6} deliveries per day in New York City.

The third and fourth rows from the top in Tables I and II display numbers based on the kinetic theory of gases and Eq. (6). Those in the bottom three rows do not depend on the kinetic theory, and so are valid for both strategies 1 and 2. All numbers in the two tables are valid for both smart and dumb drones, except for *p*_{max}, which applies only to smart drones. EPD estimates extend over a huge range: from a lower bound value of 4.7 (Table I) to an upper bound value of 170,000 (Table II). These numbers are encounters, not collisions. Even in the worst case, the number of collisions would be acceptable if the collision probability *p *≤1.58 × 10^{−7} (assuming CPD_{acc} = 2.74 × 10^{−2}). This implies a reliability, r = 1 − *p *≥* *0.9999998. For comparison, *ISO Standard* 26262 implies a reliability of seven nines for future self-driving cars for each 30-mile round trip commute in the US.^{20}

## III. A SIMPLE STRATEGY FOR REDUCING THE ENCOUNTER RATE

One way to reduce EPD values is to fly drones in the same direction and speed but at different levels. If the 100 m thick drone fly zone is divided into four strata, with, say, drones in the 75–100 m stratum flying north at the same speed, those in the 50–75 m stratum flying east, etc., there will be no collisions except for drones ascending or descending through different strata. Let the density of those drones moving vertically be α*N,* where 0 ≤ α ≤ 1. In one round trip, the average time a drone spends moving vertically is $4l/sv$, where $l=50\u2009m$ is the average distance the drone travels to reach the correct stratum, $sv$ is its vertical speed, and the factor of 4 accounts for the four vertical segments of each round trip. The average time spent in horizontal motion is $d/s$, where *d* is the round-trip distance and *s* is the horizontal speed. For typical numbers ($sv=5\u2009\u2009m/s$, $s=48\u2009km/h=13\u2009m/s,$ and $d=16\u2009km)$, $\alpha \u22480.032\u226a1,$ so nearly all of the drones are traveling horizontally.

The probability of a vertically traveling drone suffering a collision with a horizontally moving one can be estimated with the help of kinetic theory. For a collision to occur, the center-to-center distance between drones must be less than 2*a*. In time $l/sv,$ a horizontal drone sweeps out a cylindrical “collision volume” equal to $4\pi a2sl/sv$, so the probability of a collision in a single round trip is $P\u2032=16\pi a2sl/svN$. From Eq. (4), the probability of a collision within a cloud of drones moving with the same speed in random directions is $P=d/\lambda Cl=16\pi a2dN/3$, so $P\u2032/P=3sl/dsv$. Using the same numbers as above, $P\u2032/P\u22480.025$, a significant reduction in the encounter rate. Of course, travel distances will increase, offsetting some of this advantage: a drone traveling 1 km north and 1 km east will travel 2 km rather than 1.4 km “as the crow flies.”

## IV. CONCLUSIONS

While drone delivery promises many economic advantages, there are numerous quality-of-life issues to be addressed. Care must be taken to ensure pedestrians are not injured when deliveries are made to urban sidewalks or front steps, as opposed to private back yards. Table II suggests that, for a 50% drone delivery fraction, 2600–24,000 drones could simultaneously occupy the airspace over New York City. Combine the 31 drones/km^{2} in Table II's upper bound 50% case with the sound of a single drone to imagine the noise level. The bottom three rows of each table give a sense of how drone-crowded the sky will be, with inter-drone separation ranging from 1600 m down to 140 m. Migrating geese beware!

## ACKNOWLEDGMENTS

Thanks to the two anonymous referees for their comments and insights, which were crucial to improving earlier versions of this paper. And thanks to Dr. Gastone Celesia, Frank Straka, and McLouis Robinet for very helpful discussions, advice, and encouragement.

### APPENDIX: CALCULATING AVERAGE INTER-DRONE SEPARATION

Tables I and II include estimates for the average separation between nearest neighbor drones. We present here a procedure by which those estimates were calculated.

The ground surface area per drone is obtained by taking the inverse of the numbers in row 6 (“Drones/km^{2}”) of Tables I and II. Define a vertical cell extending from the bottom of the drone fly zone to the top, 100 m higher, having a square horizontal cross section of area *L*^{2} equal to the ground surface area per drone. *L* is also the horizontal distance between nearest neighbor cell centerlines. Let the single drone in each cell be on average located on the cell centerline. The average separation between nearest neighbor drones *Z*_{av} is approximately equal to the rms separation of the drones *Z*_{rms}, where

and *H *=* *100 m, *x* is the height of the first drone, and *y* is the height of a drone in an adjacent cell.