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Queueing network templates

Queueing Network — Passport Office Counter

Public counter handling 74 applicants an hour, where a two-desk supervisor referral room with only six places blocks more than a quarter of the people sent to it and quietly sheds them from the network.

Template previewQueueing network
Passport office — public counterOpen queueing network · 5 stations · 7 routes · arrivals 74/h · target ρ 85%10.890.1110.6 back74/harrivals1exitexit 0.4Ticket kioskλ 74/h · ρ 51.4%Wq 17.9s · Lq 0.37W 67.9s · L 1.4M/M/2 · P(wait) 34.9%Document check×10λ 77.7/h · ρ 77.7%Wq 58s · Lq 1.25W 7m · L 9.02M/M/10/40 · blocked 0%Biometrics booth×7λ 69.2/h · ρ 74.1%Wq 55.2s · Lq 1.06W 5.4m · L 6.25M/M/7 · P(wait) 37.1%Supervisor referralbottleneckλ 8.55/h · ρ 92.9%Wq 21.4m · Lq 2.21W 39.4m · L 4.07M/M/2/6 · blocked 27.5%Payment + issue×10λ 69.2/h · ρ 80.7%Wq 1.5m · Lq 1.78W 8.5m · L 9.84M/M/10 · P(wait) 42.5%StationModelcK1/μλλ effaρP(wait)P(block)LqWqLWTicket kioskM/M/22∞50s74/h74/h1.0351.4%34.9%0%0.3717.9s1.467.9sDocument checkM/M/10/4010406m77.7/h77.7/h7.7777.7%36%0%1.2558s9.027mBiometrics boothM/M/77∞4.5m69.2/h69.2/h5.1974.1%37.1%0%1.0655.2s6.255.4mSupervisor referralM/M/2/62618m8.55/h6.2/h2.5692.9%84.8%27.5%2.2121.4m4.0739.4mPayment + issueM/M/1010∞7m69.2/h69.2/h8.0780.7%42.5%0%1.781.5m9.848.5mBottleneck: Supervisor referral at ρ = 92.9% on 2 servers — 4 servers (2 more) would bring it under the 85% target.Offered 74/h · throughput 71.6/h · blocked 2.36/h (3.2%) · L 30.58 in system · W 25.6m end to end · 5 stationsStable — every station holds ρ below 1. Network response time is Little's law on the solved flows: W = L / throughput.

Make it your own.

title "Passport office — public counter"
target 85%
arrivals "Ticket kiosk" 74/h

station "Ticket kiosk"        { servers: 2;  service: 50s }
station "Document check"      { servers: 10; service: 6m; capacity: 40 }
station "Biometrics booth"    { servers: 7;  service: 4m30s }
station "Payment + issue"     { servers: 10; service: 7m }
# Only six people fit in the supervisor's room, desks included. Anyone who
# cannot get in is turned away and told to come back — a real loss.
station "Supervisor referral" { servers: 2;  service: 18m; capacity: 6 }

route "Ticket kiosk" -> "Document check" 1.0
route "Document check" -> "Biometrics booth" 0.89
route "Document check" -> "Supervisor referral" 0.11
route "Biometrics booth" -> "Payment + issue" 1.0
route "Payment + issue" -> exit 1.0
route "Supervisor referral" -> "Document check" 0.6   # cleared, rejoin the queue
route "Supervisor referral" -> exit 0.4