Queueing Complications
Real-world phenomena that break the textbook 1 / k assumptions: not every server can do every job, people leave the line, some jobs get priority. From ECE459 L33.
Why does this matter?
Textbook M/M/k assumes every server is interchangeable, every arriving job joins and waits, and everyone is served FIFO. Real queues (food halls, call centres, Service Ontario, theme parks) violate all three. Once models get messy enough, closed-form math gives up and you simulate.
Interchangeability
Textbook M/M/k assumes every server can deliver every service. Food hall breaks that, gelato stand can’t make tacos. Interchangeability is a spectrum:
- Total: just hungry, any cuisine works
- Partial: want tacos but would accept shawarma
- None: need a driver’s licence, a health card isn’t a substitute
Depends on both service nature and requester needs (vegetarian rules out shawarma, vegan rules out pizza).
Leaving the queue [HB13]
Three ways customers disappear without being served:
- Balking: look at the line, decide it’s too long, don’t join
- Reneging: join, give up before reaching the front
- Loss: turned away at the door because the system is at capacity. Modeled as M/M/k/k (buffer = servers, overflow requests dropped). Origin: early phone systems, if all circuits were busy, the call just failed
Balking and reneging both need a waiting-time estimate: visible queue length, posted signs, or guesswork. Estimates break when people join ahead of you (priority).
Priority
Observation: elderly person with mobility issues pushed to the front at Service Ontario. Fair, but raises wait for everyone behind. Four questions to ask about any priority scheme:
- How much does priority help the prioritized group?
- How much does it hurt the non-prioritized group?
- Can it steer people toward less-popular options?
- If everyone has priority, nobody does. What’s the saturation point?
FastPass simulation [Per21]
When closed-form math stops scaling, simulate. Disney FastPass creates a virtual queue; FastPass+ reserves slots in advance. Key findings:
- Standby waits go up (priority cuts in)
- Overall waits drop: baseline 58 min → FastPass 40 min → FastPass+ 42 min. Gains come from steering guests to less-popular attractions
- Average rides/day: 3.31 → 3.77 → 4.23
- FastPass+ increases inequality: more high-usage guests, more low-usage guests. Expert-user knowledge asymmetry compounds this in the real park
Takeaway
Complex priority systems encourage under-utilized capacity but reward system-gamers. Complexity = attack surface for capacity planning.