MARK and TABULATE Block in GPSS & Coffee Shop Simulation

Part A: MARK Block

The MARK block is used to record the current simulation clock time into a specified parameter of a transaction. This timestamp is later used to calculate the transit time (total time spent in the system).

Syntax:

MARK A

• If operand A is specified: the current clock time is stored in parameter A of the transaction
• If A is blank: the clock time is stored in the Mark Time word of the transaction
• Used in conjunction with the TABULATE block to measure time intervals
• Helps collect time-in-system statistics for performance analysis

Symbol:

+--------+
|  MARK  |
|   A    |
+--------+

Part B: TABULATE Block

The TABULATE block is used to collect statistical data and record frequency distributions into a TABLE entity. Each time a transaction passes through this block, one entry is made in the specified table.

Syntax:

TABULATE A

• Operand A specifies the TABLE name/number where data is recorded
• Works with a previously defined TABLE definition statement

TABLE Definition Syntax:

name TABLE A, B, C, D

Symbol:

+-----------+
| TABULATE  |
|     A     |
+-----------+

How MARK and TABULATE work together:

1. MARK records the arrival time in a transaction parameter (e.g., parameter 1)
2. TABULATE later computes the elapsed time as: Current Clock Time - Marked Time
3. This elapsed time is entered into the specified TABLE for statistical analysis

Part C: Coffee Shop Simulation Model

Given Data

System Type

This is an M/M/1 queuing system:

• Poisson arrivals (exponential inter-arrival times)
• Exponential service times
• Single server (one barista)
• FIFO discipline

Theoretical Performance (Pre-Simulation)

Server Utilization (rho):

rho = lambda / mu
    = (1/3) / (1/2.5)
    = 2.5 / 3
    = 0.833  (83.3%)

Expected customers served in 480 minutes:

Customers = 480 / 3 = 160 customers (approximately)

Since rho < 1, the system is stable.

Block Diagram

         +------------------+
         |   GENERATE 3     |  <-- Customers arrive every 3 min (exponential)
         +------------------+
                  |
         +------------------+
         |     MARK 1       |  <-- Record arrival time in parameter 1
         +------------------+
                  |
         +------------------+
         |   QUEUE  LINE    |  <-- Customer joins the waiting line
         +------------------+
                  |
         +------------------+
         |  SEIZE  BARISTA  |  <-- Customer occupies the barista
         +------------------+
                  |
         +------------------+
         |  DEPART  LINE    |  <-- Customer leaves the queue count
         +------------------+
                  |
         +------------------+
         |  ADVANCE 2.5,FN$ |  <-- Service time avg 2.5 min (exponential)
         +------------------+
                  |
         +------------------+
         | RELEASE BARISTA  |  <-- Barista is freed for next customer
         +------------------+
                  |
         +------------------+
         | TABULATE THRUPUT |  <-- Record transit time in table
         +------------------+
                  |
         +------------------+
         |   TERMINATE      |  <-- Customer leaves the system
         +------------------+

GPSS Program Code

* ================================================
* Coffee Shop Simulation Model
* Single Barista, 8-hour simulation (480 minutes)
* ================================================

        SIMULATE

* Define Table for Transit Time (time in system)
* MP1 = transit time from parameter 1
* Upper limit of first class = 0
* Class width = 1 minute
* Number of classes = 20

THRUPUT TABLE MP1,0,1,20

* ------------------------------------------------
* Customer Arrival and Service Segment
* ------------------------------------------------

        GENERATE  3,FN$XPDIS    ;Customers arrive, avg every 3 min (exponential)
        MARK      1              ;Store arrival time in parameter 1
        QUEUE     LINE           ;Customer joins the waiting line
        SEIZE     BARISTA        ;Customer seizes the single barista
        DEPART    LINE           ;Customer departs from queue
        ADVANCE   2.5,FN$XPDIS  ;Service takes avg 2.5 min (exponential)
        RELEASE   BARISTA        ;Barista is released
        TABULATE  THRUPUT        ;Record time-in-system in table
        TERMINATE                ;Customer exits the system

* ------------------------------------------------
* Timer Segment (controls simulation end time)
* ------------------------------------------------

        GENERATE  480            ;Generate one transaction at 480 minutes
        TERMINATE 1              ;Decrement termination counter to 0

* ------------------------------------------------
* Control Statement
* ------------------------------------------------

        START     1              ;Run simulation until counter reaches 0
        END

Explanation of Each Block

What the 8 Hour Run Produces

The line THRUPUT TABLE MP1,0,1,20 is a definition statement rather than a block: MP1 is the transit time measured from parameter 1, the first frequency class closes at 0, each class is 1 minute wide, and 20 classes are kept. Since MARK writes the arrival time and TABULATE reads it back through MP1, every entry in THRUPUT is one customer's total time in the shop.

When the timer transaction drives the termination counter to zero at 480 minutes, GPSS prints its standard report: the utilisation and number of entries of the facility BARISTA, the average and maximum contents of the queue LINE together with the average waiting time, and the THRUPUT table with its mean and standard deviation. With $\lambda = 1/3$ per minute and $\mu = 1/2.5 = 0.4$ per minute, the M/M/1 results give the values those simulated figures should settle near:

$$\rho = \frac{\lambda}{\mu} = 0.833, \qquad L_s = \frac{\rho}{1 - \rho} = 5 \text{ customers}, \qquad L_q = \frac{\rho^2}{1 - \rho} = 4.17 \text{ customers}$$

$$W_s = \frac{1}{\mu - \lambda} = 15 \text{ minutes}, \qquad W_q = \frac{\rho}{\mu - \lambda} = 12.5 \text{ minutes}$$

So over the 8 hour day about 160 customers arrive, the barista is busy for roughly 83 percent of the time, and a customer spends about 15 minutes in the shop of which about 12.5 minutes is spent waiting for service. A single 480 minute run is short, so the figures the simulation reports scatter around these analytical values rather than reproduce them exactly.

Simulation and Dynamic Physical Model: Mechanical-Electrical Analogy

1. What is Simulation?

Simulation is the imitation of the operation of a real-world process or system over time. It involves the generation of an artificial history of the system, and the observation of that artificial history to draw inferences concerning the operating characteristics of the real system that is represented.

In other words, simulation allows us to study the behavior of a system without actually building or experimenting on the real system, by creating a model that mimics its behavior.

2. Physical Model

A physical model is one in which system attributes are represented by physical measures such as voltage, and the system activities are represented by physical laws.

Physical models are of two types:

3. Dynamic Physical Model

A dynamic physical model is one which changes with time or which is a function of time. Dynamic models follow the changes over time that result from system activities.

Example: In a wind tunnel, small aircraft models are kept and air is blown over them. The wind velocity changes with time, and forces are measured with the help of transducers. Since the system changes with time, it is an example of a dynamic physical model.

4. Analogy Between Mechanical System and Electrical System

The mechanical and electrical systems are classic examples of dynamic systems. The analogy is established by identifying corresponding (analogous) quantities between the two systems.

4.1 Mechanical System

Consider a spring-mass-damper system where:

• A mass M is attached to a spring (spring constant K) and a damper (damping coefficient B)
• An external force F(t) is applied

The governing equation (Newton's second law) is:

$$M\frac{d^2x}{dt^2} + B\frac{dx}{dt} + Kx = F(t)$$

Where:

• x = displacement
• dx/dt = velocity
• d²x/dt² = acceleration

4.2 Electrical System

Consider a series RLC circuit where:

• A resistor R, inductor L, and capacitor C are connected in series
• A voltage source V(t) drives the circuit

The governing equation (Kirchhoff's Voltage Law) is:

$$L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = V(t)$$

Where:

• q = charge
• dq/dt = current (i)

4.3 Analogy Table

By comparing the two governing equations:

$$M\frac{d^2x}{dt^2} + B\frac{dx}{dt} + Kx = F(t) \quad \longleftrightarrow \quad L\frac{d^2q}{dt^2} + R\frac{dq}{dt} + \frac{1}{C}q = V(t)$$

4.4 Diagram

Mechanical System          Electrical System
                           
  F(t)                       V(t)
   |                          |
  [M] Mass              [L] Inductor
   |                          |
  [B] Damper            [R] Resistor
   |                          |
  [K] Spring            [C] Capacitor
   |                          |
  GND                        GND

5. Significance of the Analogy

• The two systems have mathematically identical governing equations, which means the behavior of one system can be studied by analyzing the other.
• This is the core idea of simulation: instead of building a costly or dangerous mechanical system, an equivalent electrical circuit can be constructed and analyzed.
• This analogy is widely used in control systems, vibration analysis, and system modeling.

Summary

Simulation imitates a real-world system over time. Dynamic physical models change with time. The mechanical system (mass-spring-damper) and the electrical system (RLC circuit) are analogous dynamic systems governed by identical differential equations, where mass corresponds to inductance, damping to resistance, spring constant to 1/capacitance, force to voltage, and displacement to charge.

Properties of Random Numbers and Auto-Correlation Test

Part 1: Properties of Random Numbers

Random numbers $R_i$ must satisfy two key properties:

1. Uniformity The numbers are uniformly distributed on $(0,1)$; every value is equally likely. The pdf is:

$$f(x) = \begin{cases} 1 & 0 \le x \le 1 \ 0 & \text{otherwise} \end{cases}$$

with $E(R) = \tfrac{1}{2}$ and $Var(R) = \tfrac{1}{12}$.

2. Independence The current value has no correlation with previous values; the probability of observing a value in a subinterval is independent of prior draws.

Part 2: Auto-Correlation Test

Given Data

• $i = 2$, $m = 3$, $N = 20$, $\alpha = 0.05$, $Z_{\alpha/2} = 1.96$

Step 1: Find M

$M$ is the largest integer such that $i + (M+1)m \le N$:

$$2 + (M+1)\cdot 3 \le 20 \Rightarrow (M+1)\cdot 3 \le 18 \Rightarrow M+1 \le 6 \Rightarrow M \le 5$$

$$\boxed{M = 5}$$

Step 2: Subsequence Values

Indices $2 + 3k$ for $k = 0,1,\dots,5$:

Step 3: Auto-Correlation Estimate

$$\hat{\rho}m = \frac{1}{M+1}\left[\sum{k=0}^{M} R_{i+km},R_{i+(k+1)m}\right] - 0.25$$

The summation runs $k = 0$ to $M$, giving $M+1 = 6$ products of consecutive pairs:

Note on convention: The standard Banks/Carson formula sums $k=0$ to $M$ using pairs $R_{i+km}\cdot R_{i+(k+1)m}$, where the last term needs $R_{i+(M+1)m} = R_{2+18} = R_{20} = 0.59$.

Including the $k=5$ term: $0.16 \times 0.59 = 0.0944$.

$$\sum_{k=0}^{5} = 0.4005 + 0.6942 + 0.4914 + 0.1827 + 0.0464 + 0.0944 = 1.9096$$

$$\hat{\rho}_m = \frac{1.9096}{5+1} - 0.25 = \frac{1.9096}{6} - 0.25 = 0.31827 - 0.25 = 0.06827$$

Step 4: Standard Deviation of the Estimate

$$\sigma_{\hat{\rho}_m} = \frac{\sqrt{13M + 7}}{12(M+1)} = \frac{\sqrt{13(5)+7}}{12(6)} = \frac{\sqrt{72}}{72} = \frac{8.4853}{72} = 0.11785$$

Step 5: Test Statistic

$$Z_0 = \frac{\hat{\rho}m}{\sigma{\hat{\rho}_m}} = \frac{0.06827}{0.11785} = 0.5793$$

Step 6: Decision

Since $|Z_0| = 0.5793 < Z_{\alpha/2} = 1.96$, we fail to reject the null hypothesis of independence.

$$\boxed{\text{The numbers are independent (no significant autocorrelation).}}$$

Markov Chain: Definition and Explanation

Definition

A Markov Chain is a sequence of random variables X₁, X₂, X₃, ... with the Markov property, namely that, given the present state, the future and past states are independent.

Formally, this can be written as:

P(Xₙ₊₁ = xₙ₊₁ | X₁ = x₁, X₂ = x₂, ..., Xₙ = xₙ) = P(Xₙ₊₁ = xₙ₊₁ | Xₙ = xₙ)

This property is called the "memoryless" property - the next state depends only on the current state, not on the sequence of states that preceded it.

Key Concepts

Example: Weather Prediction

Consider a simple weather model with two states:

• State 1: Sunny (S)
• State 2: Rainy (R)

Transition Probabilities

Suppose:

• If today is Sunny, probability of Sunny tomorrow = 0.8, Rainy = 0.2
• If today is Rainy, probability of Sunny tomorrow = 0.4, Rainy = 0.6

Transition Matrix

$$P = \begin{pmatrix} 0.8 & 0.2 \ 0.4 & 0.6 \end{pmatrix}$$

Where rows represent the current state and columns represent the next state.

State Diagram

        0.8                  0.6
    [Sunny] ←→ 0.2 →→ [Rainy]
        ↑                    ↑
        ←←←← 0.4 ←←←←←←←←←←

Applying the Markov Property

If today is Sunny, the probability that it will be Rainy after 2 days is:

• Sunny → Sunny → Rainy = 0.8 × 0.2 = 0.16
• Sunny → Rainy → Rainy = 0.2 × 0.6 = 0.12
• Total = 0.16 + 0.12 = 0.28

This calculation depends only on today's state (Sunny), not on what the weather was yesterday. This demonstrates the Markov property.

Application in Queuing Systems

Markov Chains are widely used in queuing models. For example, a single-server queue can be modeled as a continuous time Markov chain with transition matrix:

$$Q = \begin{pmatrix} -\lambda & \lambda \ \mu & -\mu \end{pmatrix}$$

The system is stable only if the utilization factor P = λ/μ < 1, meaning the arrival rate must be less than the service rate.

Summary

• Markov Chain models systems that transition between states over time.
• The key assumption is that the future depends only on the present, not the past.
• It is widely used in simulation, queuing theory, weather forecasting, and network modeling.

Iterative Process of Calibrating a Model

Definition

Calibration is the iterative process of comparing the model to the real system, making adjustments to the model, comparing again, and so on, until the model adequately represents the real system. The comparison of the model to reality is carried out by a variety of tests.

Steps in the Iterative Calibration Process

The calibration process follows a cycle of repeated steps as described below:

Step 1: Build the Initial Model

Construct an initial version of the simulation or mathematical model based on available data, assumptions, and understanding of the real system.

Step 2: Run the Model

Execute the model to generate outputs under defined conditions or inputs.

Step 3: Compare Model Output to Real System

Compare the model's output against observed real-world data or behavior. This comparison is carried out using a variety of tests, such as:

• Statistical tests (mean, variance comparison)
• Graphical comparisons (trend matching)
• Sensitivity analysis

Step 4: Identify Discrepancies

Identify where and how the model output deviates from the real system behavior. These discrepancies indicate areas where the model needs adjustment.

Step 5: Adjust the Model

Modify model parameters, assumptions, or structure to reduce the identified discrepancies. Adjustments may include:

• Changing input distributions
• Modifying rate parameters
• Revising model logic or equations

Step 6: Repeat (Iterate)

Return to Step 2 and repeat the process. The cycle continues until the model output closely matches the real system behavior within an acceptable tolerance.

Diagram of the Iterative Calibration Process

+------------------+
|  Build/Adjust    |
|     Model        |
+--------+---------+
         |
         v
+--------+---------+
|   Run the Model  |
+--------+---------+
         |
         v
+--------+---------+
| Compare Model to |
|   Real System    |
+--------+---------+
         |
    +---------+
    | Match?  |
    +---------+
    /         \
  YES          NO
   |            |
   v            v
Accept      Adjust Model
Model       (Go back to top)

Key Points

Conclusion

Calibration ensures that the simulation model is a reliable representation of the real system. Since no model is perfect on the first attempt, the iterative nature of calibration is essential to progressively improve model accuracy before it is used for decision-making or analysis.

Feedback System

Definition

A feedback system is a system in which the output of the system is measured and fed back (returned) to the input side to be compared with the desired input, so that the system can automatically adjust and control its behavior to achieve the desired output.

In other words, a feedback system uses information about its own output to regulate or modify its future behavior. It is also known as a closed-loop system.

Key Components of a Feedback System

Block Diagram

         Error
Input --> [Comparator] --> [Controller] --> [Process] --> Output
              ^                                               |
              |_____________[Feedback Path]__________________|

• Error = Input - Feedback (Output)
• If error is positive, the system increases its action.
• If error is negative, the system decreases its action.

Types of Feedback

1. Positive Feedback: The feedback signal adds to the input, amplifying the output (less common in control systems).
2. Negative Feedback: The feedback signal opposes the input, stabilizing the output (most commonly used).

Examples

Example 1: Thermostat (Temperature Control System)

• A room thermostat is set to a desired temperature (e.g., 25°C).
• A temperature sensor continuously measures the actual room temperature.
• If the actual temperature falls below 25°C, the heater is turned ON.
• If the actual temperature rises above 25°C, the heater is turned OFF.
• The measured temperature is continuously fed back to the comparator to maintain the desired level.

Example 2: Automatic Water Tank (Float Valve)

• The desired water level is the reference input.
• As water level drops, the float goes down, opening the valve to allow water in.
• As water reaches the desired level, the float rises and closes the valve.
• The water level (output) is continuously fed back to control the valve (input).

Example 3: Human Body Temperature Regulation

• The human body maintains a temperature of approximately 37°C.
• If body temperature rises, sweating occurs to cool it down.
• If body temperature falls, shivering occurs to generate heat.
• This is a natural biological feedback system.

Advantages of Feedback Systems

• Reduces error and improves accuracy
• Provides stability to the system
• Makes the system self-correcting
• Reduces the effect of disturbances and noise

Summary

A feedback system continuously monitors its output and uses that information to adjust its input, ensuring the system behaves as desired. It is widely used in engineering, biology, economics, and simulation modeling to maintain control and stability.