Hello everyone,
I've been struggling with that problem for a long time now and I would really appreciate any help/suggestions. This problem looks far from being trivial. So see it as the challenge of your day.
My problem is as follows:
I am a bidder with a bid x on a trading market and hope being matched with a seller that is willing to accept my bid. I would like to derive my expected waiting time before obtaining a seller as a function of this bid x.
I know that:
Over time, other bidders arrive on the market following a Poisson process at rate lambda_b and post a random bid price drawn from a uniform distribution [0,1].
Similarly, sellers arrive on the market following a Poisson process at rate lambda_s at rate lambda_s and post a random sell price drawn from a uniform distribution [0,1].
Note that a seller will accept to sell if the price is equal or higher than its sell price. Oppositely, a bidder will also accept to buy if the price is equal or lower than its bid price.
At the time of my match, I know that:
No bidder has a higher bid price than me (otherwise he would get the match), there is at least one seller that is willing to sell me the product for my price.
I've been struggling with that problem for a long time now and I would really appreciate any help/suggestions. This problem looks far from being trivial. So see it as the challenge of your day.
My problem is as follows:
I am a bidder with a bid x on a trading market and hope being matched with a seller that is willing to accept my bid. I would like to derive my expected waiting time before obtaining a seller as a function of this bid x.
I know that:
Over time, other bidders arrive on the market following a Poisson process at rate lambda_b and post a random bid price drawn from a uniform distribution [0,1].
Similarly, sellers arrive on the market following a Poisson process at rate lambda_s at rate lambda_s and post a random sell price drawn from a uniform distribution [0,1].
Note that a seller will accept to sell if the price is equal or higher than its sell price. Oppositely, a bidder will also accept to buy if the price is equal or lower than its bid price.
At the time of my match, I know that:
No bidder has a higher bid price than me (otherwise he would get the match), there is at least one seller that is willing to sell me the product for my price.