Spreadsheet Models for Managers

This reading is especially relevant for Session 12Service Systems
 

Service systems are systems in which customers arrive and are serviced — the airline check-in counter, the order counter of a fast-food restaurant.

This page gives some explanations of the formulas for service system performance parameters. You don’t need to read this to do the homework, but if you’re curious about where the formulas come from, read on.

• Assumptions
• Single-server systems
• Multi-server systems

Assumptions

In modeling service systems, we’ll make three simplifying assumptions.

The arrival rate is constant, and arrivals are described by a Poisson distribution with parameter l. This is just a fancy way of saying that customer arrivals are independent of each other.

The service rate is constant, and the distribution of service completion times is described by an exponential distribution with parameter m.

The system is in equilibrium — that is, the doors to the facility opened a long time ago. We must assume this because just after the doors open, there is nobody in line, and it takes a while for things to settle, which makes it a lot more difficult to describe.

From these assumptions, we’ll derive relationships for some important performance parameters of a service system. These formulas are useful in decision-making regarding capacity utilization for the system. The quantities that are most useful in describing the system are.

• L_s: the average number of customers in the system, either waiting in line or being serviced.
• W_s: the total elapsed time to get through the system, including both waiting time and service time.
• L_q: the average number of customers in the queue.
• W_q: the average elapsed time spent waiting in the queue.
• P_n: the probability that there are n customers in the system.
• P_n>k: the probability that there are more than k customers in the system.
• ρ: the average utilization rate of the server. This is the fraction of the time that the server isn’t idle.
• L_a: the average length of non-empty lines.
• W_a: the average waiting time for customers who actually have to wait.
  

Single-server systems

A most important relationship is the Fundamental Balance Equation, which is based on the equilibrium assumption. If we let P_n be the probability that there are n customers in the system, either waiting in line or being served, then

λP_n-1 = μP_n

What does this mean? Just that since we’re in equilibrium, on average the number of customers in the system can’t change. Thus if there are n-1 customers, the probability that we go to n in the next interval of time t is λtP_n-1. Similarly, if there are n customers in the system, the probability that we finish one is μtP_n. These two probabilities must be equal if we want to stay in equilibrium:

λtP_n-1 = μ t P_n

and we see that the factor of t on both sides can be eliminated.

We can use the Fundamental Balance Equation to derive the average number of customers in the system. We’ll do this by first calculating the probability P_n that there are n customers in the system. Let P₀ be the probability that there are no customers in the system and P₁ be the probability that there is one customer in the system. Then

and

Continuing in this way P_n = (λ/μ)ⁿP₀. Now since

it follows that

Here we assumed that λ < μ, in order to close the sum of powers of λ/μ. This sum is just a geometric progression. Unless this assumption is valid, the sum of powers is infinite. But this assumption is the only interesting case anyway, because if μ < λ, the waiting lines become infinite in length and never reach equilibrium. Thus P₀ = 1 - λ/μ and

From this we can immediately derive ρ, the utilization rate of the server. Since P₀ is the probability of zero customers in the system, it’s also the probability that the server is idle. Thus, the utilization rate ρ of the server is just 1 - P₀, so

Another performance measure of interest is the probability that there are fewer than j customers in the system:

Thus

Now that we know the probability of there being n customers in the system, it’s relatively simple to figure out the average number of customers in the system.

We won’t prove it, but this last form can be rewritten as

We can also derive a relationship between L_s and W_s. Let’s think about the state of the system just after a customer has been serviced. On average there are then L_s customers in the system, and they have all arrived in the time it took that last customer to get through the system. On average, the time it took that last customer to get through the system was W_s, so the average number of new customers that arrived in that time was λ W_s. Thus

Since we have already have an expression for L_s in terms of λ and μ, we have

Let’s now find expressions for the average length of the queue, and for the time spent waiting in the queue. For some businesses, these attributes of the system are critical determiners of customer satisfaction (or dissatisfaction!), and they also determine the size of the waiting facility.

Consider the state of the system just after a customer is called to be serviced. On average there are then L_q customers in the queue, and they have all arrived in the time it took that last customer to be called by the server for service. On average, the time it took that last customer to get through the queue was W_q, so the average number of new customers that arrived in that time was λ W_q. Thus

Since μ is the average number of customers serviced per unit time, the average time required to service a customer is 1/μ. And since the average total time spent in the system is equal to the average time spent waiting in line plus the average time spent being serviced,

Since we know how to express W_s in terms of λ and μ, it follows that

and

To find the average length of non-empty lines, we divide the average length of a queue by the probability that there is a queue at all:

The expected waiting time for those who actually wait is the average time to service L_a customers:

Multi-server systems

When a single service line feeds multiple servers, as in airline passenger ticketing facilities, the average service rate is just a multiple of the service rate for a single server. Suppose there are s servers in the system. Then if the average service rate for a single server is μ, the average service rate for the multiple server system is sμ.

Things are now complicated due to the numbers of different ways the servers can be occupied. Although the Fundamental Balance Equation for the multiple server system is just

The other relations we “derived” so easily for the single-server system are no longer so simple. Here they are:

Probability of no customers in the system		(svc-1)
Probability of n customers in the system		(svc-2)
Probability of n>s customers in the system		(svc-3)
Average number in the system		(svc-4)
Average number waiting		(svc-5)
Average waiting time		(svc-6)
Average time spent in the system		(svc-7)

Last Modified: Wednesday, 27-Apr-2016 04:15:26 EDT