Question7 What is the transition rate from state (0, 0, 3, 4) to state (0, 0, 4, 4)? (Four (4) marks for the correct answer. Zero (0) marks for a wrong answer or if unanswered. )Select only one of the following options: [math: 2λL] [math: 4λL] [math: λL] The starting state of the given transition is invalid. [math: 2λH] [math: 3λH] The end state of the given transition is invalid. 0 [math: 4λH] [math: λH] [math: 3λL] ResetMaximum marks: 4 Flag question undefined单项选择题
A
2 𝜆 𝐿
B
4 𝜆 𝐿
C
𝜆 𝐿
D
The starting state of the given transition is invalid.
E
2 𝜆 𝐻
F
3 𝜆 𝐻
G
The end state of the given transition is invalid.
H
0
I
4 𝜆 𝐻
J
𝜆 𝐻
K
3 𝜆 𝐿

Question

Question7 What is the transition rate from state (0, 0, 3, 4) to state (0, 0, 4, 4)? (Four (4) marks for the correct answer. Zero (0) marks for a wrong answer or if unanswered. )Select only one of the following options: [math: 2λL] [math: 4λL] [math: λL] The starting state of the given transition is invalid. [math: 2λH] [math: 3λH] The end state of the given transition is invalid. 0 [math: 4λH] [math: λH] [math: 3λL] ResetMaximum marks: 4 Flag question undefined单项选择题

A

2 𝜆 𝐿

B

4 𝜆 𝐿

C

𝜆 𝐿

D

The starting state of the given transition is invalid.

E

2 𝜆 𝐻

F

3 𝜆 𝐻

G

The end state of the given transition is invalid.

H

0

I

4 𝜆 𝐻

J

𝜆 𝐻

K

3 𝜆 𝐿

BlackTom AI · Accepted Answer

An increase in the third coordinate by one (3 → 4) is a single arrival of the H-type stream, so the transition rate equals λ_H.

1) Relevant knowledge points and concepts
- State vector (a, b, c, d) records counts in four positions; a change of +1 in one coordinate corresponds to a single arrival to that class.
- In a continuous-time Markov chain with independent arrival streams, each arrival type occurs at its own Poisson rate (λ_H for H arrivals, λ_L for L arrivals). Single-event transitions occur at the single-stream rate, not multiplied unless multiple identical independent sources contribute simultaneously.
- Simultaneous multi-customer jumps are not modeled here; transitions change one coordinate by ±1.

2) Step-by-step reasoning
- Compare the two states: (0, 0, 3, 4) → (0, 0, 4, 4). Only the third coordinate changes: 3 increases to 4; all other coordinates unchanged.
- That change corresponds to one arrival to the third-class queue/category.
- Identify which arrival stream affects the third coordinate: by model convention the third coordinate corresponds to the H-type class, so the governing arrival rate is λ_H.
- Because only a single H arrival is required, the instantaneous transition rate is λ_H.

3) Intermediate calculation steps
- Delta vector = (0,0,4,4) − (0,0,3,4) = (0,0,1,0).
- Single event of type H → rate = λ_H (no multiplication by number of servers/customers).

4) Why the correct answer is correct
- The change is exactly one H-arrival; the model assigns rate λ_H to each such arrival. There is no factor of 2,3,4 because only one arrival happens and there are no multiple identical independent sources to sum.

5) Why other options are incorrect
- Any option with λ_L is wrong because a low-class arrival would change a different coordinate (or be governed by λ_L), not the third coordinate here.
- Multiples like 2λ_H, 3λ_H, 4λ_H are incorrect because they imply simultaneous multiple arrivals or multiple independent sources contributing, which is not the case.
- “The start/end state invalid” or “0” is incorrect because both states are valid and the transition is allowed (a single H arrival is possible).