Queuing Theory. 3. Birth-Death Process. Law of Motion Flow balance equations Steady-state probabilities: , if

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1 1 Queuing Theory 3. Birth-Death Process Law of Motion Flow balance equations Steady-state probabilities: c j = λ 0λ 1...λ j 1 µ 1 µ 2...µ j π 0 = 1 1+ j=1 c j, if j=1 c j is finite. π j = c j π 0 Example 2: (page 1121) The customer service representatives of MingFan Insurance Company receive an average of 50 calls per hour. The time between calls follows an exponential distribution. A customer service representative can handle an average of 20 calls per hour. The time required to handle a call is also exponentially distributed. Ming- Fan can put up to 5 people on hold. If 5 people are on hold, a call is lost to the system. MingFan has 3 service representatives. a) What fraction of the time are all operators busy? b) What fraction of all calls is lost to the system?

2 4. M/M/1/GD/ / Traffic intensity of queuing system, ρ = λ/µ Steady-state probabilities: π 0 = 1 ρ π j = ρ j (1 ρ) If ρ 1, no steady-state exists. # of customers: L = j=0 jπ j = ρ 1 ρ L q = j=1 (j 1)π j = ρ2 1 ρ L s = 0π 0 + 1(π 1 + π ) = 1 π o = ρ Little s Law: If steady-state exists L = λw L q = λw q L s = λw s Results is independent of # of services, interarrival time distribution, service discipline, and service time distribution. Wait Times: W = 1 (µ λ) W q = λ µ(µ λ) 2

3 Example 3: (page 1128) An average of 10 cars per hour arrive at a single-server drive-in teller. Assume the average service time for each customer is 4 minutes, and both interarrival times and service time are exponential. Answer the following questions: 1) What is the probability that the teller is idle? 2) What is the average number of cars waiting in line for the teller? (A car that is being served is not considered to be waiting in line.) 3) What is the average amount of time a drive-in customer spends in the bank parking lot (including time in service)? 4) On the average, how many customers per hour will be served by the teller? Example 5: (P 1129) Machinists who work at a tool-and-die plant must check out tools from a tool center. An average of ten machinists per hour arrive seeking parts. At present, the tool center is staffed by a clerk who is paid $6 per hour and who takes an average of 5 minutes to handle each request for tools. Since each machinist produces $10 worth of goods per hour, each hour that a machinist spends at the tool center costs the company $10. The company is deciding whether or not it is worthwhile to hire (at $4 per hour) a helper for the clerk. If the helper is hired, the clerk will take an average of only 4 minutes to process requests for tools. Assume that service and interarrival times are exponential. Should the helper be hired? 3

4 5. M/M/1/GD/c/ Limited physical space ( N ) in the system - the M/M/1/GD/c/ system. If the system is full customers leave the system. We assume that these customers are forever lost. If λ µ steady-state probabilities π 0 = 1 ρ 1 ρ N+1 π j = ρ j π 0 ( j= 1, 2,..., N) π j = 0 ( j= N+1, N+2,...) L = ρ[1 (N+1)ρN +Nρ N+1 ] (1 ρ N+1 )(1 ρ) If λ = µ π j = 1 N+1 ( j= 0, 1, 2,..., N) L = N 2 L s = 0π 0 + 1(π 1 + π ) = 1 π o L q = L L s Wait Times: W q = L q λ(1 π N ) W = L λ(1 π N ) Actually arrival rate of the system? Of all people arriving only λ λπ N actually enter the system Steady-state always exists, why? 4

5 Example: Consider a shoeshine shop with 3 chairs. The shoeshine agent can serve 2 customers per minute. On average 3 customers arrive at the shoeshine stand per minute. If both times are assumed exponential, a) What fraction of all customers will enter the system? b) On the average, how many customers per minute will the agent complete the service? c) How long do entering customers wait in the system? d) How long do entering customers wait in line? e) What is the probability that the server is busy? 5

6 6. M/M/s/GD/ / A single queue, but multiple servers - the M/M/s/GD/ / system. When there are s or fewer customers in the system, no customer waits. Steady-state probabilities ρ = λ sµ c j = π 0 = 1 π j = c j π 0 (sρ) j (j = 0, 1,..., s) j! (sρ) j (j = s + 1, s + 2,...) s!s j s 1+c 1 +c = 1 s 1 (sρ) i i=0 i! + (sρ)s s!(1 ρ) The probability that the servers are busy P (j s) = (sρ)s π o s!(1 ρ) It can be shown that Lq = P (j s)ρ 1 ρ then, W q = L q λ = P (j s) sµ λ Since W s = 1 µ, we have L s = λ µ L = L q + λ µ W = L λ = L q λ + 1 µ = W q + 1 µ 6

7 For an M/M/s/F CF S/ / system, we have: P (W > t) = e µt {1 + P (j s) 1 e µt(s 1 sρ) s 1 sρ } P (W q > t) = P (j s)e sµ(1 ρ)t Note: The book provides a table giving lookup values for P (j s) for different values of and s on page Example: (Problem 3 in P1144 of the Text) In this problem, all interarrival and service times are exponential. a) At present the finance department and marketing department each have their own typists. Each typist can produce 25 letters per day. Finance requires an average of 20 letters per day and marketing requires an average of 15 letters per day. Determine for each department the average time between a letter request and its completion. b) Suppose that the two typists were grouped into a typing pool; that is, each typist would be available to type letters for either departments. For this arrangement, calculate the average time between a letter request and its completion. c) Comment on the results of part a) and b). d) Under pooled arrangement, I. what is the probability that more than day will elapse between a letter request and its completion.? II. determine the fraction of time that a particular typist is idle? 7

8 Example 8 in P The manager of a bank must determine how many tellers should work on Friday. For every minute a customer stands in line, the manager believes that a delay cost of 5 cents is incurred. An average of 2 customers per minute arrive at the bank. On the average, it takes a teller 2 minutes to complete a customer s transaction. It cost the bank $9 per hour to hire a teller. Interarrival times and service time are exponential. To minimize the sum of service costs and delay costs, how many tellers should the bank have working on Friday? 8

9 7. GI/G/ /GD/ / A single queue and an infinite number of servers. Average arrival rate: λ; average service time: 1/µ. W = 1 µ, L = λ µ If interarrival times are exponential (M/G/ /GD/ / ), π j = (λ/µ)j e λ/µ j! Example In 1995, there were approximately 380,000 births in Canada. On average, Canadians live about 78 years. What is the population of Canada? 9

10 8. M/G/1/GD/ / Interarrival times are exponentially distributed. But service times are not exponentially distributed. Pollaczek-Khinchin (PK) equation L q = λ2 σ 2 +ρ 2 2(1 ρ) where: ρ = λ/µ µ = the mean service rate. σ 2 = the variance of the variance of service process. And of course from this equation we can get : L = L q + L s = L q + ρ (since W s = 1/µ, L s = λw s = λ/µ = ρ) W q = L q /λ W = L/λ = W q + 1/µ It is difficult to determine steady-state probability for this system. Example how service time variance can greatly affect wait times. Assume a process where the arrival rate is 5 customers per unit of time and the service rate is 8 customers per unit of time. If we have an M/M/1 system, the length of the queue is: If we have a deterministic system, the length of the queue is: 10