The purpose of this section is to illustrate the use of real options in the field of queuing. Unlike traditional discussion of real options where the focus tended to be on the use of real options to understand the impact of expanding or contracting a project, I will use real options in this section to quantify the impact of expanding a queuing operation – although the impact here is on the bottom line as opposed to the value of optionality. In Chapter 4,1 discussed queuing briefly and how simulations can be used to quantify metrics of a queue. In the discussion, I also touched on the convention (proposed by Kendall in 1953) that is in the form of A/B/k/S/N/P to help classify and group different types of queuing problems. To illustrate the use of real options in queuing, I will start with the M/M/l queue and discuss standard queuing metrics (e.g., probability of having 0/1/2/3... people in the queuing system, expected waiting time experienced by any new arrival into the queuing system until the moment the service is completed). I then expand my discussion to include the M/M/2 queue following which I discuss the M/M/k queue. Since most of the results developed are asymptotic in nature, I investigate the effectiveness of these results when the queue is operational only in finite time (as most queues are in practice). I conclude the illustration with a discussion on the issue of optimally managing the queue when there are costs involved.

M/M/1 Queue

In Chapter 4,1 discussed the simulation of an M/M/l queue. In this section, I will discuss the use of analytical methods to obtain some interesting metrics associated with the queue (that was computed in Chapter 4 using simulations).

As before, I first assume that customers enter a queuing system as according to an exponential distribution with a rate of λ and get served by a server (where the service time follows an exponential distribution with rate μ). To keep my illustration simple, I further assume that the maximum number of customers served and the calling population size from which customers arrive are infinite, and whoever enters the queue first is served first (i.e., FIFO).

Probability of Having n Customers in the Queue Letting Pn (t) represent the probability of there being n people in the queuing system at time t (including the person being served) where n = 0, 1, 2, 3, ..., and t > 0, by considering the possible events that can happen over a infinitesimal time Δt, one has

(9.3a)

(9.3b)

Taking limits as, equations (9.3a) and (9.3b) simplify to

(9.4a)

(9.4b)

For the queue to reach stationarity conditions (i.e., be time independent and stabilize), one would want that (for n = 0, 1, 2, 3, ...). Doing this allows equations (9.4a) and (9.4b) to yield for n = 1, 2, 3, ..., where =.^{[1]}

Using the identity , it follows that for

Expected Waiting Tine To find the expression for the expected waiting time (the time until service is completed), one needs a waiting-time distribution. To get this, let W represent the time taken for a new arrival into the queue to finish receiving the service and f(w) be the probability density function (pdf) of W (where w > 0). Then:^{[2]}

Thus, the waiting time pdf is exponentially distributed with parameter . As a consequence the expected waiting time is

Percentage Of Downtime To calculate this quantity, it is important to first observe that the server would have nothing to do if there is no one in the queue. Hence the percentage of downtime is simply given by the expression .

As a consequence, the percentage of time the server would be busy is simply

[1] is sometimes also called queue intensity where . This condition on p is needed for the infinite series to converge and arrive at the expression .

[2] Since the time for each service has an exponential pdf with rate μ, the service time for n + 1 independent services is simply the sum of n + 1 independent exponential pdfs. As the moment-generating function of an exponential pdf with rate μ is , the moment-generating function of the sum of n + 1 independent identically distributed exponential pdf is – which is the moment-generating function of a gamma pdf with parameters n + l and /t (if X is a gamma variate with parameters n + 1 and μ, then the pdf of X has the form for x > 0).

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