14Long-Haul Wireless

When a storm crosses the line of towers between Chicago and New Jersey, the microwave links under it fade, and for as long as the rain lasts the fastest route between the futures and the stocks is fibre again, three milliseconds slower. Shkilko and Sokolov used exactly those afternoons as an experiment: weather episodes that “temporarily remove the speed advantages of the fastest traders by disrupting their microwave networks” were associated with lower adverse selection and lower trading costs. The rain gave the slower traders a few hours without the race, and the market’s liquidity improved while it lasted.

Chapter 10 measured the routes against their floors and chapter 13 took the glass apart. This chapter does the same for the air: the geometry of a line-of-sight hop, the power it has to spare, the rain that takes it away, the millimetre-wave and laser links inside a metropolitan area, the shortwave radio that crosses oceans, and what these links can carry. Its model runs a labelled storm across a chain of towers between Aurora and Carteret and counts the minutes on fibre.

14.1 Microwave links: line of sight, Fresnel zones and towers

Laughlin, Aguirre and Grundfest give the same mid-hop rules (8.7 and 1/50) and draw the consequence for the Chicago–New York networks: with towers of about 100 metres, frequencies near $6\,\mathrm{G}\mathrm{Hz}$, $K$ below 4/3 and obstacles of 10 metres, a hop over flat terrain cannot be much longer than about 70 kilometres, so a 1 200-kilometre route needs twenty hops or more. Table 14.1 reproduces the arithmetic.

The budget follows from physics once the equipment is fixed. Free space loses $92.45 + 20\log_{10} f + 20\log_{10} d$ decibels ($f$ in gigahertz, $d$ in kilometres); a dish of diameter $D$ gains $10\log_{10}[\eta(\pi D f/c_0)^2]$. With the model’s equipment (30 dBm transmitted, 1.8-metre dishes of 60% efficiency, $3\,\mathrm{dB}$ of other losses, a receiver threshold of $-70$ dBm, all assumptions), a 60-kilometre hop keeps between 31 and $54\,\mathrm{dB}$ of margin (Table 14.2). Higher frequencies gain more in the dishes than they lose in free space; the rain takes it back.

14.2 Weather, rain fade and availability

Recommendation ITU-R P.838-3 gives rain’s specific attenuation as a power law of the rain rate, $\gamma = k R^{\alpha}$ decibels per kilometre for $R$ millimetres an hour, with coefficients tabulated by frequency and polarisation. At $11\,\mathrm{G}\mathrm{Hz}$ (horizontal polarisation) $k = 0.01772$ and $\alpha = 1.2140$: a storm of $50\,\mathrm{m}\mathrm{m}/\mathrm{h}$ costs $2.05\,\mathrm{dB}$ per kilometre. At $6\,\mathrm{G}\mathrm{Hz}$, $k = 0.0007056$ and $\alpha = 1.59$; at $80\,\mathrm{G}\mathrm{Hz}$, $k = 1.1704$ and $\alpha = 0.7115$. Figure 14.2 shows the spread: in a light rain of $2.5\,\mathrm{m}\mathrm{m}/\mathrm{h}$, a factor of about 740 between 6 and $80\,\mathrm{G}\mathrm{Hz}$.

A 60-kilometre hop at $11\,\mathrm{G}\mathrm{Hz}$ keeps $36.4\,\mathrm{dB}$ of margin in the model: rain of $18.4\,\mathrm{m}\mathrm{m}/\mathrm{h}$ along its whole length would break it, but a storm cell ten kilometres across needs $80\,\mathrm{m}\mathrm{m}/\mathrm{h}$. At $6\,\mathrm{G}\mathrm{Hz}$ the same hop survives $64\,\mathrm{m}\mathrm{m}/\mathrm{h}$ everywhere; at $80\,\mathrm{G}\mathrm{Hz}$ it fails in a drizzle, which is why long hops use the lower bands and short ones can use the higher.

def gamma_db_km(rain_mm_h, f_ghz, pol="h"):
    k, a = coefficients(f_ghz, pol)
    return k * rain_mm_h ** a


def rain_db(hop, rain_mm_h, wet_km=None):
    wet = hop.d_km if wet_km is None else min(wet_km, hop.d_km)
    return gamma_db_km(rain_mm_h, hop.f_ghz, hop.pol) * wet


def critical_rain(hop, wet_km=None):
    k, a = coefficients(hop.f_ghz, hop.pol)
    wet = hop.d_km if wet_km is None else min(wet_km, hop.d_km)
    fm = fade_margin_db(hop)
    return 0.0 if fm <= 0 else (fm / (k * wet)) ** (1 / a)

Availability follows from how often the critical rate is exceeded, which is a property of the climate along the route, published in rain-rate statistics that this chapter does not use. With a labelled model in which it rains 5% of the time with a mean rate of $4\,\mathrm{m}\mathrm{m}/\mathrm{h}$ (exponentially distributed), one 59-kilometre hop at $11\,\mathrm{G}\mathrm{Hz}$ soaked end to end is down 0.047% of the year, about four hours; twenty such hops failing independently would put the route on fibre for about 0.9% of the time. The model is crude; the structure is not: availability multiplies along a chain, and a route is only as dry as its wettest hop.

14.3 Millimetre waves and lasers

Inside a metropolitan area the trade-off reverses: hops are short, rain has little path to act on, and bandwidth matters. ICE’s New Jersey routes between Mahwah and Secaucus, run with Anova Financial Networks, combine millimetre-wave and free-space optics over 5-gigabit radio channels; between Mahwah and Carteret they use several 1-gigabit channels. The two technologies fail differently (rain for millimetre waves, fog for lasers), so a route that carries both loses both only when rain and fog come together.

14.4 Shortwave radio across oceans

The geodesic from Secaucus NY4 to Slough LD4 is $5551\,\mathrm{k}\mathrm{m}$: $18.52\,\mathrm{m}\mathrm{s}$ of light in vacuum, against $27.07\,\mathrm{m}\mathrm{s}$ in straight glass and $29.48\,\mathrm{m}\mathrm{s}$ for the fastest published subsea route of chapter 13. A single reflection from a layer 300 kilometres high lengthens a 5 551-kilometre path by only about 0.6% in a flat-earth approximation, so a skywave link can beat any cable by about ten milliseconds before its equipment, with a bandwidth far below a microwave channel’s and an availability that follows the ionosphere, hour by hour.

14.5 Licences, towers, bandwidth and what is sent

A microwave route is public in a way a fibre route is not: in the United States its paths are licensed, and Laughlin, Aguirre and Grundfest used the FCC’s licence records to document at least fifteen Chicago–New York networks, their towers and their lengths. The spectrum is narrow: private networks there use mostly 30- and 40-megahertz channels in the 6- and 11-gigahertz bands, at about 140 to 190 megabits a second per channel in their account. Venue-sold services are narrower still: 1 to 10 megabits of private bandwidth on ICE’s Toronto service, 10 megabits on SIX’s Zurich–Frankfurt network.

What is sent is therefore chosen: the few fields of the few instruments that move another market (an index future’s best price, a trade), in compact formats, with everything else on fibre. The radio’s own latency matters: Laughlin and his co-authors describe radios on the market in 2010 that added tens to thousands of microseconds per hop, mostly in forward error correction and interleaving, and estimate a few to twenty microseconds per hop for the networks then being licensed. A route of twenty hops multiplies each microsecond by twenty.

14.6 A storm across the route

The model runs a storm across the Aurora–Carteret route: twenty hops of $59\,\mathrm{k}\mathrm{m}$ on the great circle ($1180.9\,\mathrm{k}\mathrm{m}$), a rain cell $30\,\mathrm{k}\mathrm{m}$ across moving across the line at $48\,\mathrm{k}\mathrm{m}/\mathrm{h}$, with its rate falling linearly from a peak at its centre to zero at its edge. While every hop holds its margin, the route runs at the published $3.982\,\mathrm{m}\mathrm{s}$; when one fails, it falls back to fibre, which the model takes as a path 20% longer than the geodesic, $6.911\,\mathrm{m}\mathrm{s}$. Everything but the geometry, the coefficients and the radio route’s published latency is an assumption.

With a peak of $100\,\mathrm{m}\mathrm{m}/\mathrm{h}$ the route spends 21 of the storm’s 38 minutes on fibre, $2.93\,\mathrm{m}\mathrm{s}$ slower; with $50\,\mathrm{m}\mathrm{m}/\mathrm{h}$, 5 minutes; with $150\,\mathrm{m}\mathrm{m}/\mathrm{h}$, 26. The same storms at $6\,\mathrm{G}\mathrm{Hz}$ cost 0, 0 and 7 minutes; at $18\,\mathrm{G}\mathrm{Hz}$, 22, 29 and 31. For the firms that race, those minutes are the afternoons of Shkilko and Sokolov’s study; for their counterparties, they are when the market is kinder.

def storm_timeline(hops, storm, radio_us, fibre_us):
    """Minute by minute while the cell crosses the route (from one radius before it to one
    after): hops down, and the latency, radio if every hop holds its margin, else fibre."""
    starts = [sum(h.d_km for h in hops[:i]) for i in range(len(hops))]
    out, m = [], 0
    while True:
        y = -storm.radius_km + storm.speed_km_min * m
        if y > storm.radius_km:
            return out
        down = 0
        half = math.sqrt(max(0.0, storm.radius_km ** 2 - y * y))
        for h, s in zip(hops, starts, strict=True):
            lo, hi = max(s, storm.at_km - half), min(s + h.d_km, storm.at_km + half)
            if hi <= lo:
                continue
            dist = math.hypot(y, (lo + hi) / 2 - storm.at_km)
            rate = storm.peak_mm_h * max(0.0, 1 - dist / storm.radius_km)
            if rain_db(h, rate, hi - lo) > fade_margin_db(h):
                down += 1
        out.append((m, down, radio_us if down == 0 else fibre_us))
        m += 1

14.7 Tutorial: a hop, the rain and a storm

14.8 Build: the radio link model

14.9 Exercises

14.10 Problem: The Storm Between Chicago and New Jersey

14.11 Interview questions