A queueing system consists of arrival streams of customers and a series of servers. When there are more customers than available servers, the remaining customers are said to wait in a queue. Customers leave the system eventually, being turned away, balking or reneging, or finishing service. Queueing theory is the theory of waiting lines. Draw the queueing model.
Queueing systems are often described by the notation A/B/s/K (originally due to Kendall), where A stands for the arrival distribution and B stands for the service distribution (D=deterministic, M =exponential (memoryless), and G=general). The interarrival times and service times are assumed to form i.i.d. se- quences that are independent of each other. The number of servers in parallel is s and K is the number
of customers the system can hold. If K is not given then it is assumed to be infinite. Unless other- wise stated, service order is assumed to be First-In-First-Out (FIFO), otherwise known as First-Come- First-Serve (FCFS). Other common service disciplines are Last-Come-First-Serve (LCFS) and Shortest Expected Processing Time (SEPT).
Examples: Since server queues (M/M/1, M/M/1/K), multiple server queues (M/M/s), and call centers
M/M/s/K). Questions:
(i.) The average number of customers in the system L (= λ ) and in the queue L(= ρ2 ), in-service L(= ρ). L = LQ + Ls.
(ii.) Average amount of time a customer spends in the system W (= 1 ) and in the queue WQ (= ρ ).
(iii.) Probability W > t (= e−(μ−λ)t) or WQ > t (= ρe−(μ−λ)t).μ−λ1−ρ
(iv.) Probability a customer does not need to wait for P0
(v.) Probability a customer finds at least 3 customers ahead of him upon arrival 1 − P0 − P1 − P2.
L = Σ nPn = (1 − ρ) Σ nρn.
Let ρn = Z. Then,n=0
|
ρ + 2ρ2 + 3ρ3 + ∙ ∙ ∙ |
= |
Z, |
|
ρ2 + 2ρ3 + ∙ ∙ ∙ |
= |
ρZ, |
|
— − − − − − − − − ρ + ρ2 + ρ3 + ∙ ∙ ∙ |
= |
— − − − (1 − ρ)Z. |
Hence, Z = ρ 2 and L = ρ
|
ρ |
0.8 |
0.85 |
0.9 |
0.95 |
0.98 |
|
L = ρ 1−ρ |
0.8 = 4 0.2 |
0.85 = 5.7 0.15 |
0.9 = 9 0.1 |
0.95 = 19 0.05 |
0.98 = 49 0.02 |
W = E(time in system|you observe n in the system upon arrival)
= Σ n + 1 ρn(1 − ρ) = 1 − ρ Σ∞ (n + 1)ρn = 1 − ρ Σ∞ nρn + Σ ρn!= 1 − ρ ρ + 1 = 1 = 1 .
μ (1 − ρ)2 1 − ρ μ(1 − ρ) μ − λ
Note that, L = λW which is the well known Little’s Law and apply to any system.
With Little’s Law, we can calculate the following using L = ρ
Example: Which cashier to hire? Cashier A is faster with mean service time of 1 min and a salary of
$10, while cashier B is slower with mean service time of 2 min and a salary of $5. Customers arrive at a rate of λ = 25 per hr. Assume it costs $0.02 (including time being served) for each minute ($1.2/hr) a customer is in the system.
If B is hired, μ = 30 per hr and ρ = 5 . W = 1 = 1 = 0.2hrs = 12min, L = λW = 25 × 0.2 = 5, Quote a fixed lead time t to all customers to achieve a service level SL. Note that Fˉ(t) is the probability that a customer stays more than t amount of time in the system. Then one should quote the shortest lead time such that Fˉ(t) = e−(μ−λ)t ≤ 1 − SL or t = − 1 ln(1 − SL).
Quote lead times based on the workload. Let tn be the lead time quoted to a customer when there are n customers in the system upon his arrival, n ≥ 0. Since the amount of time the customer will spend in the system is gamma(n + 1, μ), one should quote the shortest lead time
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