Soiled linen arrives all day, washers run a fixed 45 min cycle, dryers a fixed 60 min cycle, and trolleys hold the loads in between. Change the numbers and watch where the pile grows. Drag to orbit, pinch or wheel to zoom.
One washer, loads of exactly one batch arriving at random, nothing downstream limiting it. The simulator's mean wait is compared with the Pollaczek-Khinchine / Kingman formula and Little's law, at four loads, for fixed and for random cycle times.
Runs 30 simulated days for each number of machines with your other settings and shows when adding more stops helping.
Batch processes. A washer takes up to B kg and always runs a fixed cycle TW, so its capacity is nWB/TW kg per hour. Dryers do the same with TD. The dryer cycle (60 min) is longer than the wash cycle (45 min), so to move the same kilograms you need nD/nW ≥ TD/TW = 1.33 dryers per washer. Three washers and three dryers is a trap: the dryers fall behind.
Offered load. If linen arrives at λ kg/h and a stage can handle C kg/h while the shift is on, its load is ρ = λ/(C · shift/24). Whichever stage has the largest ρ is the bottleneck. A shorter shift cuts every capacity by the same factor and leaves the linen to wait overnight.
Waiting at one washer. For random arrivals and a service time with mean s and variability cs (standard deviation / mean), the Pollaczek-Khinchine formula for the average wait before starting is:
A fixed programme has cs = 0 (the M/D/1 queue) and waits half as long as an exponential one. Either way Wq → ∞ as ρ → 1: at 90 % load a load waits 4.5 cycles, at 99 % it waits about 50. Kingman's formula extends this to any arrival variability: Wq ≈ ρ/(1−ρ) × (ca² + cs²)/2 × s.
Little's law. In a stable system, average amount in the system = arrival rate × average time in the system: L = λ W. Here it is checked on kilograms for the whole laundry and on loads for the queue in front of the washer, with L measured by integrating over time and W from individual loads.
Trolleys and the pile-up. Each trolley holds 60 kg. If the wet trolleys are full, a finished washer must keep its load (it is blocked), cannot start another, and the soiled pile grows. The same happens behind the dryers. The stage that is slowest fills the trolleys in front of it and starves the stages after it. Bigger buffers delay the pile-up but do not raise throughput, and by Little's law they raise the time linen spends in the building.
Daily peaks. Arrivals follow a random process whose rate is λ(1 + a cos(2π(h − 10)/24)), highest at 10:00. During the peak (and the night, when nobody runs the machines) the instantaneous load exceeds 1 and a queue builds even if the daily average is fine.
Simplifications. Linen arrives in bags of about 5 kg that are merged into loads in order. A washer starts when the pile holds at least the "minimum fill" share of a batch; a dryer takes one washed load; fold/press processes a load in (kg / rate). Transport time, breakdowns, people, linen types and water/energy use are ignored. The exact formulas above apply to a single washer fed one load per arrival; with several machines and stages in series the results are measured by simulation rather than given by one closed formula, and the in-room hand-offs happen only while the shift is on.