Dynamic Pricing for a Multi-Product Make-to-Order Queue
We consider a firm that produces multiple products and operates in a monopoly market, and in that, has power to influence the demand for each product by varying its price. We study the problem of constructing an optimal state dependant pricing strategy and an associated sequencing rule to maximize system profits.
The model is a multi-class
M/M/1 queue with controllable arrival rates, linear demand curves and quadratic
holding costs incurred by the firm. Using a hierarchy of fluid and diffusion
approximations we show that for the single-product firm, linear pricing, i.e.,
where price increases linearly with the queue length, is asymptotically optimal
as the capacity of the system and the potential demand increase. In addition,
it is economically optimal to operate the system close to the heavy traffic
regime.
In the multi-product case, the asymptotically optimal pricing and sequencing controls decouple. The optimal pricing is now linear in the total workload in the system, while the optimal sequencing rule is the greedy heuristic that serves the job class that is incurring holding costs at the highest rate at any given time. Since workload fluctuates slowly compared to the queue lengths, the pricing decisions will also change at a much slower time scale than the system state, which is desirable from a practical viewpoint. Numerical results illustrate that indeed linear pricing performs very well in practice.