The Switching Matrix

From Tangled Copper Loops to Automated Exchanges

On the morning of March 10, 1891, the United States Patent Office granted Patent No. 447,918 to an undertaker in Kansas City, Missouri, named Almon Brown Strowger. The patent model sitting on Strowger's workbench had been assembled out of a cylindrical cardboard collar box, a hundred brass dressmaker pins pushed in ten circular rows of ten through the cardboard wall, a vertical metal rod fitted with a sweeping contact arm, and two electromagnets salvaged from a telegraph sounder. Strowger built the contraption after noticing that whenever a prominent citizen died in Kansas City, his mortuary phone remained silent while his cross-town competitor got the funeral contract—and discovering that the local Bell telephone switchboard operator was his competitor's wife, who was manually plugging callers asking for "an undertaker" into her husband's line.

To understand why a nineteenth-century city depended on human switchboard operators—and why Strowger's collar-box mechanism stayed at the heart of global telecommunications for nearly a century—requires looking at the combinatorial scaling law of physical wires. When Alexander Graham Bell patented the telephone in 1876, the earliest commercial lines were point-to-point metallic loops connecting two specific buildings: a factory owner's desk to his warehouse, or a doctor's home to an apothecary. If a network of $N$ subscribers wanted every telephone to be able to talk directly to every other telephone via dedicated wires, the number of physical lines $L$ scaled quadratically:

Connecting 50 telephones point-to-point required 1,225 separate wires; connecting 1,000 subscribers required 499,500 wires; connecting 10,000 subscribers required 49,995,000 lines. By introducing a central exchange office where each subscriber owned just one twisted copper pair—the "local loop"—running to a central switchboard, the wire count collapsed from $\frac{N(N-1)}{2}$ down to $N$. Even so, by 1887, Lower Manhattan's Broadway and Wall Street were darkened by ninety-foot cedar poles carrying thirty crossarms and up to 300 overhead wires per pole. When the Great Blizzard of March 12, 1888, coated New York's overhead web in heavy wet snow and whipped it with seventy-mile-per-hour gales, thousands of poles snapped in domino cascades, forcing telephone utilities to bury twisted-pair copper cables in underground clay ducts and scale up central switching offices.

Inside those central offices, human operators sitting before vertical wooden jack panels became the new bottleneck. Connecting a call across a large city required up to fifteen seconds of manual cord patching, and growing subscriber counts threatened to require a workforce exceeding the entire young adult population of major cities. Strowger eliminated the human operator by turning the caller's finger into a remote mechanical actuator. In his step-by-step rotary switch, each subscriber's telephone was wired to a central office shaft facing a semi-cylindrical bank of one hundred brass contacts arranged in ten vertical levels of ten horizontal positions. When a caller turned a rotary dial to "4" and released it, a spring governor interrupted the 48-volt direct-current loop four times at ten pulses per second. In the central office, a vertical lifting electromagnet stepped the wiper shaft upward four rows. During the brief pause before the caller dialed the second digit, a relay shifted connection to a rotary magnet; dialing "7" sent seven current pulses that stepped the wiper horizontally seven positions across the fourth row, clamping the copper contacts together to complete the circuit. By chaining multiple Strowger switches in stages, a caller could dial seven or ten digits and physically step through millions of line combinations across a continent.

Yet even with automated Strowger switches, telephone companies could not afford to lay enough inter-office trunk cables between neighborhood exchanges for every subscriber to talk simultaneously. How many shared trunk lines $C$ were needed between two exchanges so that callers rarely heard an "all trunks busy" tone? In 1917, a Danish mathematician named Agner Krarup Erlang, working in the basement laboratory of the Copenhagen Telephone Company, solved the problem by modeling random call arrivals as a memoryless Poisson process. If callers generate an average traffic intensity of $A = \lambda h$ Erlangs (where $\lambda$ is the mean call arrival rate per hour and $h$ is the average call duration in hours), the exact probability $B(C, A)$ that an arriving call finds all $C$ shared trunk lines occupied and is blocked is given by the Erlang B formula:

Erlang's formula revealed a powerful non-linear economy of scale in physical trunking. To hold blocking probability below $1\%$ ($B \le 0.01$), a small rural trunk carrying $A = 2\text{ Erlangs}$ of traffic required $C = 7\text{ copper pairs}$ (3.5 lines per Erlang), whereas a major metropolitan trunk carrying $A = 50\text{ Erlangs}$ required only $C = 64\text{ copper pairs}$ (1.28 lines per Erlang). Armed with Strowger's clattering electromechanical step switches and Erlang's probability tables, engineers wired thousands of local offices into a hierarchical continental switching matrix whose last electromechanical banks in rural North America did not click shut until the 1990s.