Working Set Window (Δ)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180

# Comprehensive analysis of window size effects
 
import random
from collections import defaultdict
 
class WindowSizeAnalyzer:
    """
    Analyze how window size affects working set behavior,
    page fault rate, and memory consumption.
    """
    
    def __init__(self, num_pages, trace):
        self.num_pages = num_pages
        self.trace = trace  # List of page accesses
        self.trace_length = len(trace)
        
    def compute_wss_for_delta(self, delta):
        """Compute average WSS for given delta over entire trace."""
        wss_values = []
        for t in range(delta, self.trace_length):
            window = self.trace[t-delta:t+1]
            ws = set(window)
            wss_values.append(len(ws))
        return sum(wss_values) / len(wss_values) if wss_values else 0
    
    def simulate_page_faults(self, delta):
        """
        Simulate page faults under working set policy with given delta.
        
        A page fault occurs when a page is accessed that is not
        in the current working set (i.e., not accessed in last delta refs).
        """
        faults = 0
        for t in range(self.trace_length):
            page = self.trace[t]
            
            # Check if page is in working set at time t
            window_start = max(0, t - delta)
            working_set = set(self.trace[window_start:t])  # Exclude current access
            
            if page not in working_set:
                faults += 1
                
        fault_rate = faults / self.trace_length
        return faults, fault_rate
    
    def analyze_window_range(self, delta_range):
        """
        Analyze behavior across a range of delta values.
        
        Returns data for plotting trade-off curves.
        """
        results = []
        
        for delta in delta_range:
            avg_wss = self.compute_wss_for_delta(delta)
            faults, fault_rate = self.simulate_page_faults(delta)
            
            results.append({
                'delta': delta,
                'avg_wss': avg_wss,
                'faults': faults,
                'fault_rate': fault_rate,
                'efficiency': (1 - fault_rate) / avg_wss if avg_wss > 0 else 0
            })
            
        return results
    
    def find_knee_point(self, results):
        """
        Find the "knee" in the trade-off curve.
        
        The knee is the point where increasing delta provides
        diminishing returns in fault rate reduction.
        """
        best_efficiency = 0
        knee_delta = results[0]['delta']
        
        for r in results:
            if r['efficiency'] > best_efficiency:
                best_efficiency = r['efficiency']
                knee_delta = r['delta']
                
        return knee_delta, best_efficiency
 
 
def demonstrate_window_effects():
    """
    Demonstrate how window size affects working set behavior.
    """
    print("Window Size Effects Analysis")
    print("=" * 70)
    
    # Generate trace with two phases
    trace = []
    
    # Phase 1: Tight loop over 10 pages
    for _ in range(500):
        trace.extend([i % 10 for i in range(10)])  # 5000 refs
    
    # Phase 2: Scattered access over 20 pages
    for _ in range(2000):
        trace.append(random.randint(0, 19))  # 2000 refs
    
    # Phase 3: Back to tight loop
    for _ in range(500):
        trace.extend([i % 10 for i in range(10)])  # 5000 refs
    
    analyzer = WindowSizeAnalyzer(num_pages=20, trace=trace)
    
    # Analyze range of delta values
    delta_range = [5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000]
    results = analyzer.analyze_window_range(delta_range)
    
    print("
{:>8} {:>10} {:>12} {:>12}".format(
        "Delta", "Avg WSS", "Fault Rate", "Efficiency"))
    print("-" * 45)
    
    for r in results:
        print("{:>8} {:>10.2f} {:>11.4f} {:>12.4f}".format(
            r['delta'], r['avg_wss'], r['fault_rate'], r['efficiency']))
    
    # Find knee point
    knee_delta, knee_eff = analyzer.find_knee_point(results)
    print(f"
Knee point: Δ = {knee_delta} (efficiency = {knee_eff:.4f})")
    print("
Interpretation:")
    print("- Very small Δ (5-10): High fault rate, small WSS")
    print("- Medium Δ (50-200): Good balance of faults and memory")
    print("- Large Δ (1000+): Minimal faults but large WSS")
    print("- The 'knee' represents optimal efficiency point")
 
 
def analyze_phase_transitions():
    """
    Analyze how window size affects adaptation to phase changes.
    """
    print("
" + "=" * 70)
    print("Window Size and Phase Transition Adaptation")
    print("=" * 70)
    
    # Create trace with abrupt phase change
    # Phase 1 (t=0-999): pages 0-4
    # Phase 2 (t=1000-1999): pages 10-14
    trace = []
    for _ in range(200):
        trace.extend([0, 1, 2, 3, 4])  # 1000 refs to pages 0-4
    for _ in range(200):
        trace.extend([10, 11, 12, 13, 14])  # 1000 refs to pages 10-14
    
    # Measure working sets at various points for different deltas
    print("
Working set contents at different times:")
    print("(Phase transition at t=1000)")
    
    for delta in [100, 500, 1000, 1500]:
        print(f"
Δ = {delta}:")
        for t in [800, 999, 1100, 1500]:
            if t < len(trace):
                window_start = max(0, t - delta)
                ws = set(trace[window_start:t+1])
                old_pages = {p for p in ws if p < 10}
                new_pages = {p for p in ws if p >= 10}
                print(f"  t={t}: old={sorted(old_pages)}, new={sorted(new_pages)}")
    
    print("
Observations:")
    print("- Small Δ adapts quickly but may lose useful old pages prematurely")
    print("- Large Δ retains old pages longer, slowing adaptation")
    print("- During transitions, WSS temporarily includes BOTH phase's pages")
 
 
# Run demonstrations
if __name__ == "__main__":
    demonstrate_window_effects()
    analyze_phase_transitions()

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211

# Adaptive window size selection strategies
 
class AdaptiveWindowManager:
    """
    Manages window size adaptively based on observed behavior.
    
    Implements multiple strategies for dynamic Δ adjustment.
    """
    
    def __init__(self, initial_delta, min_delta=10, max_delta=10000):
        self.delta = initial_delta
        self.min_delta = min_delta
        self.max_delta = max_delta
        
        # Tracking state
        self.recent_faults = []
        self.recent_wss = []
        self.adjustment_interval = 100  # Adjust every N references
        self.refs_since_adjust = 0
        
        # Tuning parameters
        self.target_fault_rate = 0.01  # Target 1% fault rate
        self.fault_rate_tolerance = 0.005  # +/- 0.5%
        
    def record_reference(self, was_fault, current_wss):
        """Record a memory reference and potentially adjust delta."""
        self.recent_faults.append(1 if was_fault else 0)
        self.recent_wss.append(current_wss)
        self.refs_since_adjust += 1
        
        # Keep bounded history
        max_history = self.adjustment_interval * 10
        if len(self.recent_faults) > max_history:
            self.recent_faults = self.recent_faults[-max_history:]
            self.recent_wss = self.recent_wss[-max_history:]
        
        # Periodic adjustment
        if self.refs_since_adjust >= self.adjustment_interval:
            self._adjust_delta()
            self.refs_since_adjust = 0
            
    def _adjust_delta(self):
        """
        Adjust delta based on recent behavior.
        
        Uses fault rate as primary signal for adjustment.
        """
        if len(self.recent_faults) < self.adjustment_interval:
            return
            
        # Compute recent fault rate
        recent = self.recent_faults[-self.adjustment_interval:]
        fault_rate = sum(recent) / len(recent)
        
        # Adjustment logic
        old_delta = self.delta
        
        if fault_rate > self.target_fault_rate + self.fault_rate_tolerance:
            # Fault rate too high: increase delta to capture more pages
            increase_factor = 1 + (fault_rate - self.target_fault_rate) * 10
            self.delta = min(int(self.delta * increase_factor), self.max_delta)
            
        elif fault_rate < self.target_fault_rate - self.fault_rate_tolerance:
            # Fault rate very low: might be wasting memory, decrease delta
            # Be more conservative about decreasing (prefer faults to thrashing)
            decrease_factor = 0.95  # Only 5% decrease at a time
            self.delta = max(int(self.delta * decrease_factor), self.min_delta)
            
        if self.delta != old_delta:
            print(f"Adjusted Δ: {old_delta} → {self.delta} (fault_rate={fault_rate:.4f})")
            
    def get_current_delta(self):
        return self.delta
 
 
class PerProcessWindowManager:
    """
    Manage separate window sizes for each process.
    
    Recognizes that different processes have different
    locality characteristics and need different Δ values.
    """
    
    def __init__(self, default_delta=1000):
        self.default_delta = default_delta
        self.process_managers = {}  # pid -> AdaptiveWindowManager
        
    def get_delta_for_process(self, pid):
        """Get current delta for a specific process."""
        if pid not in self.process_managers:
            self.process_managers[pid] = AdaptiveWindowManager(
                initial_delta=self.default_delta
            )
        return self.process_managers[pid].get_current_delta()
    
    def record_reference(self, pid, was_fault, current_wss):
        """Record reference for a specific process."""
        if pid not in self.process_managers:
            self.process_managers[pid] = AdaptiveWindowManager(
                initial_delta=self.default_delta
            )
        self.process_managers[pid].record_reference(was_fault, current_wss)
        
    def get_statistics(self):
        """Get per-process delta statistics."""
        stats = {}
        for pid, manager in self.process_managers.items():
            stats[pid] = {
                'delta': manager.get_current_delta(),
                'fault_history_len': len(manager.recent_faults)
            }
        return stats
 
 
class PhaseAwareWindowManager:
    """
    Manage window size with awareness of execution phases.
    
    Detects phase transitions and adjusts behavior accordingly.
    """
    
    def __init__(self, initial_delta=1000):
        self.delta = initial_delta
        self.base_delta = initial_delta
        
        # Phase detection state
        self.recent_pages = []
        self.phase_window = 100  # Look at last N accesses for phase detection
        self.previous_phase_signature = set()
        self.in_transition = False
        self.transition_start = 0
        
    def record_page_access(self, page, time):
        """Record page access and detect phase transitions."""
        self.recent_pages.append(page)
        
        if len(self.recent_pages) > self.phase_window * 2:
            self.recent_pages = self.recent_pages[-self.phase_window * 2:]
        
        if len(self.recent_pages) >= self.phase_window:
            self._detect_phase_transition(time)
            
    def _detect_phase_transition(self, current_time):
        """Detect if we're transitioning between phases."""
        current_signature = set(self.recent_pages[-self.phase_window:])
        
        if self.previous_phase_signature:
            # Jaccard similarity between current and previous phase
            intersection = len(current_signature & self.previous_phase_signature)
            union = len(current_signature | self.previous_phase_signature)
            similarity = intersection / union if union > 0 else 0
            
            if similarity < 0.3:  # Less than 30% overlap = phase transition
                if not self.in_transition:
                    print(f"Phase transition detected at time {current_time}")
                    self.in_transition = True
                    self.transition_start = current_time
                    # Temporarily increase delta during transition
                    self.delta = int(self.base_delta * 1.5)
                    
            elif similarity > 0.8 and self.in_transition:
                # Phase has stabilized
                print(f"Phase stabilized at time {current_time}")
                self.in_transition = False
                self.delta = self.base_delta
                
        self.previous_phase_signature = current_signature
        
    def get_current_delta(self):
        return self.delta
 
 
# Demonstration of adaptive approaches
def demonstrate_adaptive_window():
    print("Adaptive Window Size Demonstration")
    print("=" * 70)
    
    manager = AdaptiveWindowManager(initial_delta=100)
    
    # Simulate workload with changing characteristics
    # Phase 1: Good locality, low faults
    print("
Phase 1: Good locality (simulated low fault rate)")
    for i in range(500):
        was_fault = (i % 50 == 0)  # ~2% fault rate
        manager.record_reference(was_fault, current_wss=10)
    
    print(f"Delta after Phase 1: {manager.get_current_delta()}")
    
    # Phase 2: Poor locality, high faults  
    print("
Phase 2: Poor locality (simulated high fault rate)")
    for i in range(500):
        was_fault = (i % 10 == 0)  # ~10% fault rate
        manager.record_reference(was_fault, current_wss=20)
    
    print(f"Delta after Phase 2: {manager.get_current_delta()}")
    
    # Phase 3: Return to good locality
    print("
Phase 3: Return to good locality")
    for i in range(500):
        was_fault = (i % 100 == 0)  # ~1% fault rate
        manager.record_reference(was_fault, current_wss=10)
    
    print(f"Delta after Phase 3: {manager.get_current_delta()}")
    
    print("
Key insight: Delta adapts to maintain target fault rate")
    print("- High faults → increase delta → larger working set → fewer faults")
    print("- Low faults → decrease delta → smaller working set → save memory")

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145

# Modeling system-wide effects of window size
 
class SystemSimulator:
    """
    Simulate system-wide effects of window size choices.
    
    Models multiple processes competing for memory with
    different window size policies.
    """
    
    def __init__(self, total_frames, num_processes):
        self.total_frames = total_frames
        self.num_processes = num_processes
        
        # Per-process state
        self.process_wss = {}  # pid -> current wss
        self.process_faults = {}  # pid -> fault count
        self.active_processes = set()
        self.suspended_processes = set()
        
    def simulate_with_delta(self, delta, num_refs, locality_strength=0.8):
        """
        Simulate system behavior with given delta.
        
        Args:
            delta: Working set window (same for all processes)
            num_refs: Number of references to simulate
            locality_strength: 0-1, higher = more locality (easier workload)
            
        Returns dict with system-wide metrics.
        """
        # Initialize processes
        pages_per_process = 100
        for pid in range(self.num_processes):
            self.process_wss[pid] = 0
            self.process_faults[pid] = 0
            self.active_processes.add(pid)
            
        # Simulate memory accesses
        total_faults = 0
        total_active_time = 0
        history = {pid: [] for pid in range(self.num_processes)}
        
        import random
        
        for ref in range(num_refs):
            # Round-robin process selection (simplified)
            pid = ref % self.num_processes
            
            if pid in self.suspended_processes:
                continue
                
            total_active_time += 1
            
            # Generate page access with given locality
            if history[pid] and random.random() < locality_strength:
                # Access a page from recent history
                if len(history[pid]) > delta // 10:
                    page = random.choice(history[pid][-delta//10:])
                else:
                    page = random.choice(history[pid]) if history[pid] else random.randint(0, pages_per_process-1)
            else:
                # Random page access
                page = random.randint(0, pages_per_process - 1)
            
            history[pid].append(page)
            if len(history[pid]) > delta * 2:
                history[pid] = history[pid][-delta * 2:]
            
            # Compute working set
            ws = set(history[pid][-delta:]) if len(history[pid]) >= delta else set(history[pid])
            self.process_wss[pid] = len(ws)
            
            # Check if access is a fault (not in working set before this access)
            prior_ws = set(history[pid][-delta:-1]) if len(history[pid]) > 1 else set()
            if page not in prior_ws:
                total_faults += 1
                self.process_faults[pid] += 1
            
            # Check for thrashing (sum of WSS > memory)
            total_wss = sum(self.process_wss.values())
            if total_wss > self.total_frames:
                # Suspend process with largest WSS
                victim = max(self.active_processes, key=lambda p: self.process_wss[p])
                self.active_processes.remove(victim)
                self.suspended_processes.add(victim)
        
        # Compute metrics
        fault_rate = total_faults / total_active_time if total_active_time > 0 else 0
        avg_active = sum(1 for pid in range(self.num_processes) if pid in self.active_processes)
        avg_wss = sum(self.process_wss.values()) / len(self.process_wss)
        
        return {
            'delta': delta,
            'fault_rate': fault_rate,
            'avg_wss': avg_wss,
            'total_wss': sum(self.process_wss.values()),
            'active_processes': len(self.active_processes),
            'suspended_processes': len(self.suspended_processes),
            'memory_utilization': min(sum(self.process_wss.values()) / self.total_frames, 1.0),
            'throughput_estimate': (1 - fault_rate) * len(self.active_processes)
        }
 
 
def analyze_system_wide_effects():
    """
    Analyze how window size affects system-wide behavior.
    """
    print("System-Wide Effects of Window Size")
    print("=" * 70)
    
    print("
Configuration:")
    print("  Total memory frames: 500")
    print("  Number of processes: 20")
    print("  Pages per process: 100")
    print("  Workload locality: 80%")
    print()
    
    # Test different delta values
    delta_values = [10, 25, 50, 100, 200, 500, 1000]
    
    print("{:>8} {:>10} {:>10} {:>10} {:>12} {:>12}".format(
        "Delta", "Fault%", "Avg WSS", "Total WSS", "Active", "Throughput"))
    print("-" * 65)
    
    for delta in delta_values:
        sim = SystemSimulator(total_frames=500, num_processes=20)
        result = sim.simulate_with_delta(delta, num_refs=10000, locality_strength=0.8)
        
        print("{:>8} {:>9.2%} {:>10.1f} {:>10.0f} {:>12} {:>12.1f}".format(
            result['delta'],
            result['fault_rate'],
            result['avg_wss'],
            result['total_wss'],
            result['active_processes'],
            result['throughput_estimate']
        ))
    
    print("
Observations:")
    print("- Larger delta → larger WSS → fewer processes can be active")
    print("- Smaller delta → smaller WSS → more processes active but higher faults")
    print("- Throughput peaks at intermediate delta (balance of processes vs faults)")
    print("- Total WSS exceeding memory → process suspension (thrashing prevention)")

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177

/*
 * Practical Working Set Window Implementation
 * 
 * This demonstrates time-based working set tracking using
 * the reference bit aging technique common in real OS kernels.
 */
 
#include <stdint.h>
#include <stdbool.h>
 
#define MAX_PAGES 4096
#define WINDOW_INTERVALS 8  // Track over 8 timer intervals
 
typedef struct {
    uint8_t age;           // Bit vector: bit i = accessed in interval (now - i)
    bool reference_bit;    // Hardware reference bit (set by MMU on access)
    bool modified_bit;     // Hardware modified bit
    uint64_t frame_number; // Physical frame if resident
    bool present;          // Currently in memory
} PageTableEntry;
 
typedef struct {
    PageTableEntry pages[MAX_PAGES];
    int num_pages;
    int wss_estimate;      // Current working set size estimate
    
    // Statistics
    uint64_t total_faults;
    uint64_t total_accesses;
} ProcessMemoryState;
 
/*
 * Update working set tracking on timer interrupt.
 * 
 * Called at regular intervals (e.g., 10ms) by timer handler.
 * This is O(num_pages) but amortized over many instructions.
 */
void timer_tick_update(ProcessMemoryState *state) {
    int working_set_size = 0;
    
    for (int i = 0; i < state->num_pages; i++) {
        PageTableEntry *pte = &state->pages[i];
        
        // Shift age bits right (age all pages)
        pte->age = pte->age >> 1;
        
        // Add reference bit as new MSB
        if (pte->reference_bit) {
            pte->age |= (1 << (WINDOW_INTERVALS - 1));
        }
        
        // Clear reference bit for next interval
        pte->reference_bit = false;
        
        // Page is in working set if ANY bit is set
        if (pte->age != 0) {
            working_set_size++;
        }
    }
    
    state->wss_estimate = working_set_size;
}
 
/*
 * Check if page is in working set.
 * O(1) - just check if age bits are non-zero.
 */
bool is_in_working_set(ProcessMemoryState *state, int page_num) {
    if (page_num < 0 || page_num >= state->num_pages) {
        return false;
    }
    return state->pages[page_num].age != 0;
}
 
/*
 * Get relative age of a page (for replacement decisions).
 * Lower value = more recently used.
 */
int get_page_age(ProcessMemoryState *state, int page_num) {
    if (page_num < 0 || page_num >= state->num_pages) {
        return WINDOW_INTERVALS + 1;  // "Very old"
    }
    
    uint8_t age = state->pages[page_num].age;
    
    // Count leading zeros to find most recent access
    if (age == 0) return WINDOW_INTERVALS + 1;
    
    int recency = 0;
    while ((age & (1 << (WINDOW_INTERVALS - 1))) == 0) {
        age <<= 1;
        recency++;
    }
    return recency;
}
 
/*
 * Select victim page for replacement.
 * Returns page NOT in working set with oldest age.
 */
int select_victim_page(ProcessMemoryState *state) {
    int best_victim = -1;
    int best_age = -1;
    
    for (int i = 0; i < state->num_pages; i++) {
        PageTableEntry *pte = &state->pages[i];
        
        // Only consider present pages
        if (!pte->present) continue;
        
        // Prefer pages NOT in working set
        if (pte->age == 0) {
            // Not in working set - immediate candidate
            // Among these, prefer unmodified (clean) pages
            if (!pte->modified_bit) {
                return i;  // Best case: not in WS and clean
            }
            if (best_victim < 0) {
                best_victim = i;
            }
        }
    }
    
    // If all pages are in working set, fall back to LRU approximation
    if (best_victim < 0) {
        for (int i = 0; i < state->num_pages; i++) {
            PageTableEntry *pte = &state->pages[i];
            if (!pte->present) continue;
            
            int age = get_page_age(state, i);
            if (age > best_age) {
                best_age = age;
                best_victim = i;
            }
        }
    }
    
    return best_victim;
}
 
/*
 * Adjust effective window size dynamically.
 * 
 * This modifies how we interpret the age bits to
 * effectively change the window without re-tracking.
 */
int compute_wss_with_effective_window(ProcessMemoryState *state, int effective_intervals) {
    if (effective_intervals > WINDOW_INTERVALS) {
        effective_intervals = WINDOW_INTERVALS;
    }
    
    // Create mask for desired number of intervals
    uint8_t mask = (1 << effective_intervals) - 1;
    mask <<= (WINDOW_INTERVALS - effective_intervals);
    
    int wss = 0;
    for (int i = 0; i < state->num_pages; i++) {
        if ((state->pages[i].age & mask) != 0) {
            wss++;
        }
    }
    return wss;
}
 
/*
 * Example: Log working set dynamics for analysis.
 */
void log_wss_dynamics(ProcessMemoryState *state, uint64_t timestamp) {
    // Could write to trace buffer for offline analysis
    // Format: timestamp, wss, fault_rate, ...
    
    double recent_fault_rate = (double)state->total_faults / 
                               (state->total_accesses + 1);
                               
    // In real implementation, would log to kernel trace buffer
    // or update /proc filesystem statistics
}