We have a model where prices remain constant throughout a single time unit, typically an hour. In this context, prices are constrained to a limited set of values.

The price levels under consideration span from H3-H1 (high 3 - high 1) to M3-M1 (medium 3 - medium 1) and L3-L1 (low 3 - low 1), where H3 represents the highest and L1 denotes the lowest price level. When the price assumes one of the values L1, L2, or L3, it is categorized as being in the low regime.

Transitioning from one level to another is only possible when the time unit increases by 1, occurring at full hours. T

H3 H2 H1 M3 M2 M1 L3 L2 L1

H3 0.5 0.20 0.15 0.10 0.05 0.0 0.0 0.0 0.0

H2 0.1 0.20 0.25 0.20 0.15 0.08 0.0 0.02 0.0

H1 0.05 0.1 0.2 0.3 0.2 0.1 0.03 0.02 0.0

M3 0.08 0.1 0.1 0.35 0.2 0.1 0.04 0.02 0.01

M2 0.05 0.02 0.05 0.15 0.5 0.15 0.05 0.02 0.01

M1 0.04 0.02 0.03 0.06 0.2 0.45 0.1 0.08 0.02

L3 0.03 0.0 0.02 0.05 0.15 0.25 0.3 0.15 0.05

L2 0.02 0.0 0.01 0.02 0.05 0.2 0.3 0.3 0.1

L1 0.01 0.0 0.01 0.03 0.05 0.1 0.2 0.3 0.3

Price 500 350 250 175 130 100 50 10 -30

in e/MwH

The processing time for a market order is modeled using a memoryless distribution with an average of 1/25 time units.

The previously outlined model lacks sufficient accuracy, as the processing system's speed increases in tandem with the system load. This escalation occurs as more computer resources are diverted from other tasks to mitigate the system load. Despite the continued sequential processing of orders, each new order, up to a total of 4 orders, introduces a new handling process to facilitate order processing.

After a certain period, this model loses accuracy as the increasing number of orders being processed tends to elevate the rate of incoming orders. In addition to the background arrival rate of , there is a linear augmentation in the generation of new orders. Given the current presence of orders in the system, there is a probability of ( + ) + () of having one new order arriving in a time interval of length .

To adapt to this scenario, the handling system undergoes modification, allowing the system's capacity to increase linearly with the load. Consequently, the probability of processing an order in

a time interval of length becomes + () for all . The likelihood of more than one request arriving or being processed is considered negligible in comparison to .

NOW compute both the mean and variance of the number of orders present in the system and present all calculations.