The above comments are correct, but it should also be noted that the sodium does not actually react with the aluminium oxide arc tube to any great extent. Wikipedia is completely wrong in this respect. The sodium does react slightly with some dopants in the crystalline structure of the alumina, notably the magnesium oxide and calcium oxide components which are present at the grain boundaries. This gradually makes the arc tube wall more porous during life. Sodium atoms are then able to diffuse along the grain boundaries and leak into the outer bulb, where they react with the glass bulb and cause additional darkening which reduces luminous flux and traps more heat inside the lamp, further exacerbating the voltage rise caused by arc tube blackening.
There is also extensive reaction between sodium and the glassy frit-seals at the arc tube ends, and with the barium tungstate emitter coating of the electrodes. In that case the sodium does not leave the arc tube but becomes chemically bound with those components.
As Dez mentioned, the primary ageing mechanism of standard HPS lamps is the changing amalgam ratio. HPS lamps reach peak efficiency and luminous flux when the delta-lambda distance between the two spectral peaks either side of the sodium resonance radiation is about 120 Angstroms. But as the sodium-mercury ratio increases and the arc tube temperature rises due to blackening, the red-wing of the sodium spectrum is broadened and the d-line is also broadened. More red and infrared radiation is produced by the plasma, so the luminous flux must decrease.
Note that these failure mechanisms are entirely different in the higher performance Unsaturated Vapour HPS lamps. These are actually characterised by a falling lamp voltage during life, since there is no excess of sodium and the whole dose is vaporised. Therefore no longer so dependent on the cold spot temperature and its changes during life as a result of blackening. USV lamps are only feasible when the sodium-sinks are reduced or eliminated. So the alumina arc tubes tend to be doped with zirconium or erbium oxides in addition to magnesium oxide, and with reduced calcium oxide impurity, which reduces sodium reactions at the grain boundaries and hence the rate of sodium loss. Of far more importance though is the change of electrode emitter material, usually one of the biggest sodium sinks. The traditional barium tungstate emitter is changed to something less reactive - in the case of the Sylvania lamps that was BSY2, barium strontium yttrate combined with a special sintering procedure. Some companies also used more resistant frit sealing glasses. Incidentally just before the USV lamps fail and the last of their sodium is consumed, there is a sudden rapid rise in voltage back up to something close to the original level. This characteristic ensures that there is no end-of-life cycling. Lamps turn completely blue when the sodium is gone, and continue to burn as a pure mercury discharge. But then of course the plasma temperature increases, and since the ceramic arc tube is not chemically stable enough to withstand a pure mercury arc (sodium or metal halides are required to protect it), the ceramic eventually disintegrates and leads to complete lamp failure.
If you want to learn more about this I can highly reccomend the book of one of my former colleagues, Sjef de Groot at Philips Eindhoven, who wrote "The High Pressure Sodium Lamp". There was a copy going cheaply on Ebay for a long time that seems not to have sold but now I cannot find it. That book is rather old and does not cover the newer developments like the USV, deluxe/white, retrofit, high xenon pressure, mercury-free lamps etc. The subject was brought just about fully up to date in a comprehensive IEEE paper written in 1993 by my old boss at Sylvania, Rudy Geens, and our American colleague Elliot Wyner. See
https://digital-library.theiet.org/doi/abs/10.1049/ip-a-3.1993.0070 I can send a copy if you do not manage to find it via the usual scientific literature sources.