| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modelling and Hypothesis Testing |
| Type | Queueing and service simulation |
| Difficulty | Moderate -0.8 This is a standard D1 simulation question requiring systematic application of given rules. Part (i) is straightforward random number allocation, part (ii) is mechanical calculation of arrival times. The queueing logic with fixed rules and provided tables makes this easier than average—it's methodical bookkeeping rather than problem-solving or requiring mathematical insight. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general |
| Time interval (seconds) | 2 | 5 | 15 | 25 |
| Probability | \(\frac { 3 } { 7 }\) | \(\frac { 2 } { 7 }\) | \(\frac { 1 } { 7 }\) | \(\frac { 1 } { 7 }\) |
\multirow{2}{*}{
| red start time | 14 | 50 | 105 | 155 | ||
| \cline { 2 - 6 } | red end time | 29 | 65 | 120 | 170 | ||
\multirow{2}{*}{
| red start time | 10 | 55 | 105 | 150 | ||
| \cline { 2 - 6 } | red end time | 25 | 70 | 120 | 165 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| e.g. 00–41 \(\rightarrow\) 2; 42–69 \(\rightarrow\) 5; 70–83 \(\rightarrow\) 15; 84–97 \(\rightarrow\) 25; 98–99 \(\rightarrow\) ignore | M1 | Some ignored |
| Correct proportions | A1 | |
| Efficient allocation | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 2, 2, 25, 15, 2 / 2, 4, 29, 44, 46 | M1 A1 | Applying rule |
| M1 A1 | Accumulating |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Full simulation table with arrival/departure times at lights 1 and 2 | M1 | |
| Departure times at light 1 correct (1 error allowed) | A1 | |
| Arrival times at light 2 correct (1 error allowed) | A1 | |
| Departure times at light 2 correct (1 error allowed) | A1 | |
| All correct | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Mean delay at first lights = 2.4 seconds | M1 A1 | cao |
| Mean delay at second lights = 4.1 seconds | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| More repetitions (cars) | B1 |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| e.g. 00–41 $\rightarrow$ 2; 42–69 $\rightarrow$ 5; 70–83 $\rightarrow$ 15; 84–97 $\rightarrow$ 25; 98–99 $\rightarrow$ ignore | M1 | Some ignored |
| Correct proportions | A1 | |
| Efficient allocation | A1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| 2, 2, 25, 15, 2 / 2, 4, 29, 44, 46 | M1 A1 | Applying rule |
| | M1 A1 | Accumulating |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Full simulation table with arrival/departure times at lights 1 and 2 | M1 | |
| Departure times at light 1 correct (1 error allowed) | A1 | |
| Arrival times at light 2 correct (1 error allowed) | A1 | |
| Departure times at light 2 correct (1 error allowed) | A1 | |
| All correct | A1 | cao |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Mean delay at first lights = 2.4 seconds | M1 A1 | cao |
| Mean delay at second lights = 4.1 seconds | A1 | cao |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| More repetitions (cars) | B1 | |5 There is an insert for use in parts (iii) and (iv) of this question.\\
This question concerns the simulation of cars passing through two sets of pedestrian controlled traffic lights. The time intervals between cars arriving at the first set of lights are distributed according to Table 5.1.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Time interval (seconds) & 2 & 5 & 15 & 25 \\
\hline
Probability & $\frac { 3 } { 7 }$ & $\frac { 2 } { 7 }$ & $\frac { 1 } { 7 }$ & $\frac { 1 } { 7 }$ \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 5.1}
\end{center}
\end{table}
(i) Give an efficient rule for using two-digit random numbers to simulate arrival intervals.\\
(ii) Use two-digit random numbers from the list below to simulate the arrival times of five cars at the first lights. The first car arrives at the time given by the first arrival interval.
Random numbers: $24,01,99,89,77,19,58,42$
The two sets of traffic lights are 23 seconds driving time apart. Moving cars are always at least 2 seconds apart. If there is a queue at a set of lights, then when the red light ends the first car in the queue moves off immediately, the second car 2 seconds later, the third 2 seconds after that, etc.
In this simple model there is to be no consideration of accelerations or decelerations, and the lights are either red or green.
Table 5.2 shows the times when the lights are red.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | l | c | c | c | c | }
\hline
\multirow{2}{*}{\begin{tabular}{ l }
first set \\
of lights \\
\end{tabular}} & red start time & 14 & 50 & 105 & 155 \\
\cline { 2 - 6 }
& red end time & 29 & 65 & 120 & 170 \\
\hline
\multirow{2}{*}{\begin{tabular}{ l }
second set \\
of lights \\
\end{tabular}} & red start time & 10 & 55 & 105 & 150 \\
\cline { 2 - 6 }
& red end time & 25 & 70 & 120 & 165 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 5.2}
\end{center}
\end{table}
(iii) Complete the table in the insert to simulate the passage of 10 cars through both sets of traffic lights. Use the arrival times given there.\\
(iv) Find the mean delay experienced by these cars in passing through each set of lights.\\
(v) How could the output from this simulation model be made more reliable?
\hfill \mbox{\textit{OCR MEI D1 2005 Q5 [16]}}