I am going to develop a queueing model in which riders and drivers arrive with inter-arrival time exponentially distributed.
All the riders and drivers arriving in the system will wait for some amount of time until being matched.
The matching process will pair one rider with one driver and takes place every Δt unit of time (i.e., Δt, 2Δt, 3Δt, ⋯). Whichever side outnumbers the other, its exceeding portion will remain in the queue for the next round of matching.
The service follows the first come first served principle, and how they are matched in particular is not in the scope of this problem and will not affect the queue modelling.
I tried to formulate it as a double-ended queue, where state indicates the exceeding number in the system.
However, this formulation didn't incorporate the factor Δt in it, it is thereby not in a batch service fashion. I have no clue how I can formulate this Δt (somewhat like a buffer) into this model.
To Guoyang Qin
Can you please explain in more detail the process of matching. The enclosed picture is the standard random walk with two competing waiting exponentially distributed independent times: If wins the 1 type - we go one step to the right, in the oposite case -we go to the left. No service is sketched. Thus as much as possible precise description of the service is needed. Now, the main doubt is caused by lack of the interpretation of negative positions: Isn't it the difference of the numbers of arrived riders and drivers?
Also, writing these words:
GQ: "The matching process will pair one rider with one driver and takes place every Δt unit of time (i.e., Δt, 2Δt, 3Δt, ⋯). Whichever side outnumbers the other, its exceeding portion will remain in the queue for the next round of matching. "
it is not explained what are the "outnumbers". My English is too weak to understand the context. Can this be explained in more simple wards like this:
The state is characterized by current values of two numbers: of riders and the drivers in the waiting room. At the instant of matching k\tmes delta t the numbers are becomming less by the miniumum f the two (hence one becomes zero) . . . .
Note that this s a kind of guess what was meant by you!
Joachim
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