Service Engineering

Recitation 5, Part 1 : Queue Build-Up Diagram

Process Flow: A supermarket receives from suppliers 300 tons of fishover the course of a full year, which averages out to 25 tons per month.The average quantity of fish held in freezer storage is 16.5 tons.On average, how long does a ton of fish remain in freezer storage betweenthe time it is received and the time it is sent to the sales department?

W = L/λ = 16.5/25 = 0.66 months, on average, is the period that a tonof fish spends in the freezer.How does one get L = 16.5? This comes out of the following inventorybuild-up diagram by calculating the area below the graph:

Inventory/Queue Build-up Diagram.

Thus, the company hires an average of 194.6 + 144 = 338.6 new employees permonth, or equivalently, 338.6× 12 = 4063.2 new employees per year.

4063.2

36, 000= 0.1128× 100% = 11.28% labor turnover during a year

= 2.82% turnover during a 3-month period

(compared with 40% at fast food, for example

and about 100% in many Call Centers).

10. Process Flow: A supermarket receives from suppliers 300 tons of fish over thecourse of a full year, which averages out to 25 tons per month. The average quantityof fish held in freezer storage is 16.5 tons.On average, how long does a ton of fish remain in freezer storage between the timeit is received and the time it is sent to the sales department?

W = L/λ = 16.5/25 = 0.66 months, on average, is the period that a ton of fishspends in the freezer.How does one get L = 16.5? This comes out of the following inventory build-updiagram by calculating the area below the graph:

Inventory/Queue Build-up Diagram.

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time (months)

inve

nto

ry L

(t)

17× 412

+ 24× 412

+ 12× 212

+ 5× 212

= 16.5

7

17× 412 + 24× 4

12 + 12× 212 + 5× 2

12 =

173 + 8 + 2 + 5

6 = 16.5.

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Service Engineering

Recitation 5, Part 2: Fluid Models.

Deterministic view

• λ(t) - instantaneous arrival rate at time t

• c(t) - instantaneous capacity of the system, at time t (maximal po-tential processing rate)

• δ(t) - instantaneous processing rate at time t. Holds δ(t) ≤ c(t).

• Q(t) - total amount of material in the system, (being processed +queued) at time t.

Assume that Q(0), λ(t) and δ(t) are given for all t ∈ [0, T ].

Then Q(t) is the solution of the nonlinear differential equation

d

dtQ(t) = λ(t)− δ(t), Q(0) = Q0, t ∈ [0, T ].

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Solving the Equation

The general solution may be very complicated.

How to ”create” or plot Q(t) = (Q(0), Q(t1), Q(t2), ..., Q(T ))?

Start with Q(0). Then for tn = tn−1 + ∆t, n = 1, 2, ...

Q(tn) = Q(tn−1) + λ(tn−1) ·∆t− δ(tn−1) ·∆t.

EXCEL software can be used.

Queueing Models. Staffing.

Each little tube (on the picture above) can be viewed as server.

Consider a queueing system with one class of customers and one servicestation.

• λ(t) - instantaneous arrival rate at time t

• µ - service rate (service capacity of each server), constant in time.

• N(t) - number of servers in the system at time t

• Q(t) - total number of customers in the system, (being served +queued) at time t.

The above implies

• c(t) = µ ·N(t)

• δ(t) = µ · (Q(t) ∧N(t)), where a ∧ b = min(a, b)

• Thus

d

dtQ(t) = λ(t)− µ · (Q(t) ∧N(t)), Q(0) = Q0, t ∈ [0, T ].

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Adding the possibility to abandon queue

• θ - abandonment rate for each customer in queue.

• The equation for Q transforms to

d

dtQ(t) = λ(t)− µ · (Q(t) ∧N(t))− θ · (Q(t)−N(t))

+, Q(0) = Q0.

Example.

Tele-SHOP is a commercial channel dealing with online sales, which isoperated by a call-center.

In order to increase profits it was decided to place a short (30 sec) dailyadvertisement on national TV, at 20:00.

It turns out that the arrivals to the call-center are well approximated by aninhomogeneous Poisson process, with an arrival rate function (customersper hour) given by the figure below:

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ate

Figure 1: Arrival rate function (customers per hour).

• The call center operates from 16:00-24:00.

• The service duration of each incoming call has an average of 12 min-utes.

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• The number of servers in the system is fixed and equals N .

• Assume that no abandonment takes place.

Plot Q(t) for N = 0, 150, 200, .

We use an Excel spreadsheet

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Dynamics of Q(t) for N = 150

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number in system Staffing

Dynamics of Q(t) for N = 200

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On the second picture, (for N = 200), Q actually satisfies

d

dtQ(t) = λ(t)− µ ·Q(t), Q(0) = 0.

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Dynamics of Q(t) for N = 0

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number in system Staffing

On the last picture, (for N = 0), Q(t) actually satisfies

Q(t) =∫ t

0λ(t)dt.

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Now assume that waiting customers can abandon from the system and theabandonment rate equals θ = 6 (i.e. each waiting customer abandons afteran average of 1/θ hours, if he was not admitted to service before).

Plot Q(t) for N = 150.

Dynamics of Q(t) for N = 150 with abandonment

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Introducing costs. Profit maximization.

Define also

• c - staffing cost rate 1.

• r - service completion reward per customer.

• s - abandonment penalty per customer.

• h - waiting cost rate 2

The total profit is

C(N) =∫ T

0[rµ(Qt ∧Nt)− (sθ + h)(Qt −Nt)

+ − cNt]dt.

It depends on the staffing function N = {N(t), 0 ≤ t ≤ T}.

How to choose the optimal staffing N = N(t) in order to

maximize the profit C(N) of running the call center?

Exact solution is usually not available.

Excel Solver can be used in order to compute an approximate solution.

1The units of c are monetary units per one unit of work (salary).2The units of h are monetary units, eg. shekels, per unit of waiting time, i.e.

shekelswaiting customer×time unit .

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Numerical examples.

1. Minimize the cost of running the call center by assuming N fixed and

• an hour work of a server costs 48 shekel, i.e. c = 0.8.

• minute waiting of a customer costs 1 shekel, i.e. h = 1.

• no abandonments (θ = 0) and no rewards (r = 0).

Solution: The optimal staffing N ∗ = 149 and the cost CN∗= 63, 534.

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number in system Staffing

2. In addition, assume that the staffing can be changed at each hour.Solution : The optimal cost CN∗

= 23, 133.

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V-model

Assume that there are two classes of customers:

• VIP (class 1), with arrival rate λ1(t),

• Regular(class 2), with arrival rate λ2(t),

• Each server serves both classes, with rates µi for class i, (i = 1, 2)

• Define Qi(t) to be the total number of class - i customers in the system,i = 1, 2.

1 2

1 2

N

Figure 2: The structure of the system.

Let Q(t) = Q1(t) +Q2(t). Assume Q(0) = 0.

• Is it possible to write the differential equation for Q(t) without anyadditional assumptions?

Answer: No, since the routing policy is not specified.

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A Routing Policy for V-model

Assume that the call center works in the preemptive-resume regime:

• at every moment a service to a customer can be interrupted (in thiscase a customer goes back to queue of its class) and resumed at a latertime.

• VIP customers are high priority customers, which means that no reg-ular customer can be in service while VIP customer is waiting.

Write the differential equation for Qi(t), i = 1, 2.

Answer:

dQ1

dt= λ1(t)− µ1(Q1 ∧N), Q1(0) = 0,

dQ2

dt= λ2(t)− µ2(Q2 ∧ (N −Q1)

+), Q2(0) = 0.

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