Dynamic Pricing To Control M/M/K Queue (By Qi Zhao)

** Parameter Explanation:**

Capacity:

The total capacity of the system which includes the queue and server.

**Original Arrival Rate:**

**The maximum arrival rate, the arrival process is assumed to be Poission process with controllable arrival rarte.**

**Service rate:**

**It is a fixed rate, and the servive times are also assumed to be independent Poission process.**

**Reserve price:**

**It is the maximum price that customers want to pay(in this model, we assume Re>100).**

**Holding cost:**

** is the penalty for the customer waitting time in the system. **

**Penalty :**

** There are two kind of penaltes namely, linear and quadratic. we can choose any one of these two method according to the real situation.**

**Discount factor :**

**The factor is used to caculate the optimal discounted value along infinite time horizon.**
** Result Analysis:**

1.The optimal price is a non-decreasing function of the number of customers in the system.

2. The optimal prices decrease when the service rate of the system increases. Alternatively, we can treat it as that the optimal prices decrease when the number of servers with same servie rate in the system increases.

3.Compared with linear penalty with same holding cost coefficient, the quadratic penalty of the holding cost will cause optimal price go to its up bound quickly.

4.There always exists a value gap between the optimal discounted value and non-optimal discounted value(the optimal value is greater). Since the limitation of length of y axis, we may can not tell the gap from the figure when the values become large.
** Imperfection in this simulation:**

In this simulation, I can't get the realistic data to simulate it. So, I use some parameters I defined to simulate the program and it turns out that total expected discounted value may go to negative for some parameters which means the company will lose some money. I know it is not a good result for a manager, but how to better define the parameter to obtain the profit depends on the realistic situation in our daily life. I need to do some investigate to get these data to do simulation more precisely.