In fact WiFi (technically standard 802.11) and WiMAX (802.16) don't compete for broadband users or applications today. That's partly because WiFi is widely deployed and WiMAX is still largely an unfulfilled promise and partly because the two protocols were designed for very different situations. However, if WiMAX is eventually widely deployed, there will be competition between them as last mile technologies.
Some people describe the difference between WiFi and WiMAX as analogous to the difference between a cordless phone and a mobile phone. Wifi, like a cordless phone, is primarily used to provide a connection within a limited area like a home or an office. WiMAX is used (or planned to be used) to provide broadband connectivity from some central location to most locations inside or outside within its service radius as well as to people passing through in cars. Just like mobile phone service, there are likely to be WiMAX dead spots within buildings.
From a techie POV, the analogy is apt at another level: WiFi, like cordless phones, operates in unlicensed spectrum (in fact cordless phones and WiFi can interfere with each other in the pitiful swatch of spectrum that's been allocated to them). There are some implementations of WiMAX for unlicensed spectrum but most WiMAX development has been done on radios which operate on frequencies whose use requires a license.
Some more subversive types (they're subversive so I can't link to them) say that WiMAX is what you get when bellheads (not a nice term) try to reinvent WiFi the way they'd like it to be. It's true that WiMAX is much more a command and control protocol than WiFi. Oversimplified, in a WiFi environment every device within reach of an access point shouts for attention whenever it's got something to transmit. In that chaos, some signals tromp on other signals; the more powerful devices and those closer to the access point tend to get more than their share of airtime like the obnoxious kid who always has his hand up in the front of the class. In WiMAX devices contend for initial attention but then are assigned times when they may ask to speak. The protocol allows the operator more control over the quality of service provided — bellheads like control.
But it's not clear that more control means better service than contentious chaos (I'm talking about technology but the same may apply to economies or bodies politic). The Internet and its routing algorithms are chaotic; the routers just throw away packets if they get to busy to handle them. Bellheads (and even smart people like Bob Metcalfe) were sure that design or lack thereof wouldn't scale. They were wrong.
Same people said that voice would never work over the Internet — there's no guarantee of quality, you see. They were wrong although it's taken awhile to prove it. Now HD voice is available on the Internet but NOT on the traditional phone network (although it could be).
Lovers of an orderly environment and those who like to keep order were absolutely sure that WiFi couldn't work once it became popular. Not only is it chaotic; it also operates in the uncontrolled environment of unlicensed frequencies along with cordless phones, bluetooth headsets, walkie-talkies and the occasional leaky microwave oven. But somehow it's become near indispensable even in places where a city block full of access points contend for the scarce frequencies.
Net: I'm not convinced that WiMAX won't suffer from its own orderliness. Did you ever fume leaving an event when an amateur cop (or a professional one) managed traffic into an endless snarl? Fact is cars at low speed usually merge better without help than otherwise. Turns out that control comes at the expense of wasted capacity. The reason that the Internet or WiFi radios can work is that the computing power necessary to deal with chaos from the edge of the network is far cheaper and less subject to disruption or misallocation than the computing power (and communication) for central command and control.
WiMAX may be too well-controlled for its own good. Moreover, if it is used only in regulated spectrum where most frequencies are idle most of the time AND licenses for the frequencies have to be purchased, it will be even less efficient than if it could contend for unlicensed spectrum.
By the way, WiFi CAN operate at distances as great as WiMAX but there are two reasons why it doesn't. One reason is that radios operating in the unlicensed frequencies are not allowed to be as powerful as those operated with licenses; less power means less distance. These regulations are based on the dated assumption that devices can't regulate themselves — but the assumption MAY be correct over great enough distances. The second reason why WiFi access points don't serve as wide an area as WiMAX access points are planned to do is the engineering belief that the problem of everybody shouting at once, even if it's surmountable in a classroom, would be catastrophic in a larger arena. Maybe.
New licensed spectrum is being made available for WiMAX and other technologies NOT including WiFi — for example, the valuable 700MHz frequencies currently used by analog over the air TV. WiMAX could have a good run because it is allowed to operate in that efficient spectrum while WiFi will eventually run out of the pitifully little spectrum that's been allocated to it. That's policy and politics and not engineering but could still be a reason for WiMAX success.
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