Chapter 5: Special Types of Queuing Models

Transcription

1 Chapter 5: Special Types of Queuing Models Some General Queueing Models Discouraged Arrivals Impatient Arrivals Bulk Service and Bulk Arrivals OR37-Dr.Khalid Al-Nowibet 1

2 5.1 General Queueing Models 1. Interarrival time : Could be General distribution Given E[inerarrival time] SD[inerarrival time]. Service Time: Could be General distribution Given E[Service time] SD[Service time] 3. System size is infinite or finite Very Complex OR37-Dr.Khalid Al-Nowibet

3 5. M/G/1 Queueing Model Characteristics 1. Interarrival time is exponential with rate λ Arrival process is Poisson Process with rate λ. Service time has a any distribution E[Service] E s is known µ 1/E s Variance of service time σ 3. Single Server 4. System size is infinite 5. Queue Discipline : FCFS Notation M / G / 1 / / FCFS OR37-Dr.Khalid Al-Nowibet 3

4 5. M/G/1 Queueing Model Steady-State Distribution State of the system system is in state n if there are n customers in the system Service Time is not Exponential Memoryless Property No Balance Equations OR37-Dr.Khalid Al-Nowibet 4

5 5. M/G/1 Queueing Model Performance Measures In steady state Given : λ, E s, σ L s L q + λe s Know 1 measure all measures are known W s W q + E s L s λw s L q λw q How to get measures???? OR37-Dr.Khalid Al-Nowibet 5

6 5. M/G/1 Queueing Model Performance Measures Pollaczek and Kinchin Formula Given : λ, E s, σ L q λ σ + ( λes) (1 λe ) s Use relations to find : L s, W s and W q OR37-Dr.Khalid Al-Nowibet 6

7 5. M/G/1 Queueing Model Verify P-K Formula: Let service time is exponential with rate µ E[S] E s 1/µ Var[S] σ 1/µ L q λ σ + ( λes) (1 λe ) s λ (1/ µ ) + λ (1/ µ ) (1 λ(1/ µ )) ( λ / µ ) (1 + ( λ / µ ) ( λ / µ )) ρ + ρ (1 ρ) ρ (1 ρ) ρ (1 ρ) OR37-Dr.Khalid Al-Nowibet 7

8 5. M/G/1 Queueing Model Verify P-K Formula: Let service time is exponential with rate µ E[S] E s 1/µ Var[S] σ 1/µ L q ρ (1 ρ) L s L q + λe s ρ (1 ρ) + λ(1/ µ ) ρ (1 ρ) + ρ ρ + ρ(1 ρ) (1 ρ) ρ + ρ ρ (1 ρ) (1 ρ ρ ) OR37-Dr.Khalid Al-Nowibet 8

9 5. M/G/1 Queueing Model Example An average of 15 cars per hour arrive according to a Poisson process at a drive-in to fast food with single window. Assume the service time is uniformly distributed between [,4] minutes. Answer the following questions in steady-state: 1. What is the average number of cars waiting in line for window?. What is the average time a customer spends to get his meal? Arrival : λ 15 cars/hour 0.5 car/min Poisson Process Service : S ~ U(,4) E s (4+)/ 3 min Var(S) (4 ) / Single window M/G/1 Queueing system OR37-Dr.Khalid Al-Nowibet 9

10 5. M/G/1 Queueing Model Example 1. Average number of cars waiting in line for window L q L q λ σ + ( λes) (1 λe ) (0.01) + (0.565) 0.5 s (0.065)(0.3333) + (0.5(3)) (1 0.5(3)) Average time a customer spends to get his meal W s W s L s / λ (L q +λe s )/λ ( (3))/ min OR37-Dr.Khalid Al-Nowibet 10

11 5.3 M/G/s/s Queueing Model Characteristics 1. Interarrival time is exponential with rate λ Arrival process is Poisson Process with rate λ. Service time has a any distribution E[Service] E s is known µ 1/E s 3. Multiple Servers s 4. System size is finite s Notation M / G / s / s / FCFS 5. Queue Discipline : FCFS OR37-Dr.Khalid Al-Nowibet 11

12 5.3 M/G/s/s Queueing Model Steady-State Distribution State of the system system is in state n if there are n customers in the system Steady state distribution exists: P n s n 0 n ρ n! ρ n n! n 0, 1,,, s Erlang Distribution OR37-Dr.Khalid Al-Nowibet 1

13 5.3 M/G/s/s Queueing Model Performance Measures 1. Blocking Probability BP: BP Pr{system is full} P s Erlang Loss Formula. Effective Arrival Rate λ e : n 0 λ e λ. (1 BP) λ. (1 P s ) 3. Average Customers in System L s : L s ρ(1 P s ) s s ρ s! n ρ n! OR37-Dr.Khalid Al-Nowibet 13

14 5.4 G/G/ Queueing Model Characteristics 1. Interarrival time has any distribution E[interarrival time] E a is known λ 1/ E A. Service time has a any distribution E[Service] E s is known µ 1/E s 3. Multiple Servers : s 4. System size is infinite Notation G / G / / / FCFS 5. Queue Discipline : FCFS OR37-Dr.Khalid Al-Nowibet 14

15 5.4 G/G/ Queueing Model Steady-State Distribution State of the system system is in state n if there are n customers in the system Steady state distribution exists: ρ λ/µ P n ρ n! n ρ e n 0, 1,,, s Performance Measure L s ρ λ/µ Poisson Distribution OR37-Dr.Khalid Al-Nowibet 15

16 5.5 Discouraged Arrivals Characteristics 1. Interarrival time is exponential with rate λ. Service time has is exponential with rate µ 3. Multiple Servers : s 4. System size is finite k 5. Queue Discipline : FCFS Customers decide not to join the queue as state increases OR37-Dr.Khalid Al-Nowibet 16

17 5.5 Discouraged Arrivals Steady-State Distribution State of the system state n number of customers in the system n 0, 1,, 3, Probability a customer enters the system when he finds n in the system is α n ; 0 < α < 1 OR37-Dr.Khalid Al-Nowibet 17

18 5.5 Discouraged Arrivals Steady-State Distribution Consider: Number of servers s identical servers System size k 5 Arrival rate λ Service rate µ Entering probability if n in the system α n (0<α<1) OR37-Dr.Khalid Al-Nowibet 18

19 5.5 Discouraged Arrivals Steady-State Distribution Balance Equations: λ αλ α λ α 3 λ α 4 λ µ µ µ µ µ cut-1 λp 0 µp 1 P 1 (λ/µ)p 0 P 1 ρp 0 cut- αλp 1 µp P (αλ/µ)p 1 P (αρ /)P 0 cut-3 α λp µp 3 P 3 (α λ/µ)p P 3 (α ρ 3 /4)P 0 cut-4 α 3 λp 3 µp 4 P 4 (α 3 λ/µ)p 3 P 4 (α 6 ρ 4 /8)P 0 cut-5 α 4 λp 4 µp 5 P 5 (α 4 λ/µ)p 4 P 5 (α 10 ρ 5 /16)P 0 P n 1 n OR37-Dr.Khalid Al-Nowibet 19

20 5.5 Discouraged Arrivals Example Consider an exit on a highway that makes cars move from the highway to the local traffic. At most 5 cars can lineup in the exit. The first car in line takes an exponential time with mean 0.5 min to leave the exit to local traffic. On average, number of cars request the exit 5 cars/min. Any arriving car to the exit find n cars present will enter the exit with probability 0.8 n. in the exit Any car finds 5 cars in the exit will not wait. Assume Poisson arrivals. Find: 1. Probability that the exit is empty?. Average number of cars enters the exit per hour? 3. Average number of cars in the exit? 4. Average time until a car enters the local traffic? 5. Probability that you cant enter the exit? OR37-Dr.Khalid Al-Nowibet 0

21 5.5 Discouraged Arrivals Example Arrivals : λ 5 cars/min Poisson Process Prob. of Discouraging α 0.8 Service: E[S] 0.5 min Exponential µ cars/min Poisson Process λ 0.8λ 0.64 λ 0.51λ λ µ µ µ µ µ OR37-Dr.Khalid Al-Nowibet 1

22 5.5 Discouraged Arrivals Example λ 0.8λ 0.64 λ 0.51λ λ µ µ µ µ µ P 1 ρp 0 (.5)P 0 P (αρ )P 0 (5)P 0 P 3 (α ρ 3 )P 0 (8)P 0 P 4 (α 6 ρ 4 )P 0 (10.4)P 0 P 5 (α 10 ρ 5 )P 0 (10.485)P 0 OR37-Dr.Khalid Al-Nowibet

23 5.5 Discouraged Arrivals Example n Sum T n P n np n Probability that the exit is empty: P 0 [ ] 1 1/ Average number of cars enters the exit per hour: λ e (hr) 60(λP 0 + αλp 1 + α λp + α 3 λp 3 + α 4 λp 4 ) 60λ(P 0 + α P 1 + α P + α 3 P 3 + α 4 P 4 ) 60 (5) (0.389) car/hr OR37-Dr.Khalid Al-Nowibet 3

24 5.5 Discouraged Arrivals Example 3. Average number of cars in the exit: L s Σ np n 3.49 cars 4. Average time until a car enters the local traffic: W s L s /λ e 3.49/ hr 1.8 min. 5. Probability that you cant enter the exit: Pr{exit is full} P OR37-Dr.Khalid Al-Nowibet 4

25 5.6 Impatient Arrivals Characteristics 1. Interarrival time is exponential with rate λ. Service time has is exponential with rate µ 3. Multiple Servers : s 4. System size is finite k 5. Queue Discipline : FCFS Customers leave the queue without service after waiting random amount of time OR37-Dr.Khalid Al-Nowibet 5

26 5.6 Impatient Arrivals Steady-State Distribution State of the system state n number of customers in the system n 0, 1,, 3, Customer leave the system without service after waiting in queue for exponential time with mean 1/β. Rate of customers leaving the queue without service β OR37-Dr.Khalid Al-Nowibet 6

27 5.6 Impatient Arrivals Steady-State Distribution Consider: Number of servers s 3 identical servers System size k 6 Arrival rate λ Service rate µ Rate of impatient customers β OR37-Dr.Khalid Al-Nowibet 7

28 5.6 Impatient Arrivals Steady-State Distribution Balance Equations: λ λ λ λ λ λ µ µ 3µ 3µ+β 3µ+β 3µ+β cut-1 λp 0 µp 1 P 1 (λ/µ)p 0 cut- λp 1 µp P (λ/µ)p 1 cut-3 λp 3µP 3 P 3 (λ/3µ)p cut-4 λp 3 (3µ+β)P 4 P 4 λ/(3µ+β)p 3 cut-5 λp 4 (3µ+β)P 5 P 5 λ/(3µ+β)p 4 cut-6 λp 5 (3µ+β)P 6 P 6 λ/(3µ+β)p 5 P n 1 n OR37-Dr.Khalid Al-Nowibet 8