4.7 Spatially Distributed Queues And The M/G/1 Queueing System

As we have already mentioned at the beginning of this chapter, many queueing systems in the urban environment do not fit neatly into the classical mold of a physically stationary server where prospective "customers" arrive and queue up until they receive service. For most of the emergency urban services-where "arrivals of customers" are perceived through telephone calls (or other means of telecommunication) from various locations in a citythe only place at which a "queue" can be identified is on the emergency system's dispatcher's desk (or in a computer's memory), where records indicate the time, origin, and nature of a succession of requests for service. The server, then, be it a person or a vehicle, must travel to the location of these incidents to provide the required service. In such cases we have a spatially distributed queue.
    In this section we shall discuss the simplest possible type of spatially distributed queue in which a single server has sole responsibility for a given district. This discussion will also motivate our derivation of some important results for M/G/1 queueing systems.

Example 1: Ambulance Service to an Emergency Medical Facility

Consider the case pictured in Figure 4.10: an emergency medical facility (EMF) is located at the center (the point where the diagonals intersect) of a rectangular district with dimensions X0 Y0 miles. The EMF has a single ambulance vehicle associated with it, which is dispatched to emergency patients and transports them to the EMF. The ambulance, when idle, is always located at the EMF.

fig.4.10.gif (13489 bytes)

    Directions of travel are parallel to the boundaries of the district and the effective travel speeds are vx. and vy (constants) in the x and y directions, as shown in Figure 4.10.
    In this district, incidents (each corresponding to one emergency patient) occur as a Poisson process in time at a rate of A per hour. Incident locations are independent of each other and are uniformly distributed over the rectangular area. Every time an incident occurs, the ambulance, if at the EMF, is immediately dispatched to the location of the call, picks up the patient, and returns to the EMF. If the ambulance is away, calls queue up and are processed in a FIFO order. We shall assume for now that no calls are ever lost, no matter how long they have to wait. We shall also assume that the pdf for the time, Z, that the ambulance spends at the location of each call for service is known and is given by fz(z) with an expectation Z and a variance sigma2.gif (521 bytes)z.
    The service time, S, in this system clearly consists of a   "travel-time" component and a "time on the scene" component. Assuming that effective travel speeds are identical on both the EMF-to-incident and the incident-toEMF portions of each trip, we have

form4.59.gif (2530 bytes)

where Tx = Dx/vx, and Ty = Dy/vy, and Dx and Dy are defined as the distances along the x and the y axes, respectively, from (to) the EMF to (from) the location of an incident.
    With the techniques presented in Chapter 3, it is an easy matter to obtain the expected value of S, which for consistency with our queueing theory notation we shall denote as 1/mu.gif (300 bytes):

form4.60.gif (21632 bytes)

    If fz(z), as we have already assumed, is known, it is also possible, at least in principle, to obtain an expression for fs(s), the service-time pdf. The derivation, however, even for the simple situation described here, promises to be tedious and time-consuming and will be omitted. It suffices to note that fs(s) cannot be a negative exponential pdf [unless fz(z) is negative exponential and the expected time on the scene is much larger than the average round-trip time, in which case it can be argued that 1/mu.gif (300 bytes) approx.gif (57 bytes) Z and, therefore, that fs(s) is approximately negative exponential as well 7]. It is therefore clear that results derived for the single-server queueing systems we have seen so far (M/M/1) do not apply in this case, since the service-time distribution at hand is a "general" one.

We shall now proceed to derive several important results for this M/G/1 queueing system. Interestingly, as we shall see, most of the results require no knowledge of the service-time distribution, fs(s), other than its mean, 1/mu.gif (300
bytes), and variance a2s.gif (1044
bytes).

    Recall first that the current state of M/M/1 (or M/M/m) queueing systems is fully described by a single item of information, the number of users (i.e., of calls in our EMF example) currently in the system. Knowledge of this number is sufficient to describe all the past history of the queueing system, as far as the future is concerned. For instance, if it is known that at some time instant, t, there are exactly n (> 0) calls in a M/M/1 system (with one call receiving service and the other n - 1 calls in the waiting line), then we can immediately state that the probability thatin the next delta.gif (317 bytes)t a service will be completed is equal to mu.gif (300
bytes)delta.gif (317
bytes)t-independently of what else has happened in the past at that M/M/1 system. For M/G/1 systems, however, this probability also depends on how long ago service began to the call that is currently receiving service. Thus, a complete description of the current state of a M/G/1 system requires, in general, specification of the values of two random variables, the number of calls currently in the system, and the time since the current service began, the latter of which is a continuous random variable. These complications make the mathematical analysis of M/G/1 systems more difficult than that of M/M/1 or M/M/m systems.
    Of the several different approaches that have been developed, the simplest one uses the trick of focusing attention on certain specific instants in time, known as epochs, when knowledge of only the number of calls currently in the queueing system is sufficient to specify its current state. Those instants of time are the times of completion of a service by the server (i.e., the instants after the ambulance has returned to the EMF and delivered a patient, in the case of our example).
    Let us then indicate these time instants as t1, t2, t3 . . . with ti, representing the instant when service to the ith patient to be transported to the EMF (beginning with some arbitrary time t = 0) is completed. A specific example for a hypothetical M/G/1 queueing system is shown in Figure 4.11.

fig4.11.gif (19236 bytes)

We can then define:

N = number of calls in the queueing system (i.e., the number of patients/incidents waiting for service) just after the instant ti-1 when service to the (i - 1)th patient is completed

R = number of new calls that arrive at the queueing system during the service time of the next patient to receive service (patient i)

N' = number of calls in the queueing system just after ti, the instant of completion of service to patient i

    Note that, by definition, R includes only those calls that arrive at the queueing system after service to the next patient (patient i) has started. This is important for the cases when, upon completion of a service, the queueing system is left empty (i.e., no more calls left to serve). For instance, the value of R for the time interval between t5 and t6, is 0 (not 1) for the sample case shown in Figure 4.11.
    The following relationship exists between the random variables N, R, and N':

form4.62.gif (3813 bytes)

The probability sigma.gif (280 bytes)r, that exactly r calls arrive during a service time is given by

sigma.gif (280 bytes)r = P{number of new arrivals during a service time= r} = pR(r)

form4.63.gif (4480 bytes)

where, as above, fs(s) represents the pdf for the service time. For any given pdf f,(s), it is then possible to determine the probabilities sigma.gif (280
bytes)r.

Exercise 4.5 How is (4.63) justified?

Hint: Given that a service lasted exactly a time t, what is the  probability that r new calls arrived during that service time?

For r = 0, 1, 2,. . . , we have

form4.64.gif (10537 bytes)

So (4.64) and (4.65) give the state-transition probabilities for successive epochs for the M/G/1 system. The state-transition diagram for our M/G/1 system, at the epochs is now shown in Figure 4.12. A state is defined by

fig4.12.GIF (21024 bytes)

"the number of calls present at the time when a service is completed." Note that the state-transition diagram of Figure 4.12 is no longer of the birth-and-death type.
    In the analysis that follows, we shall be interested only in the expected values Lbar.gif (304 bytes), Wbar.gif (557 bytes), Lbar.gif (304 bytes), and Wbar.gif (557 bytes)q, which can be derived without using the state-transition diagram. We shall, therefore, ignore Figure 4.12 from here on.
    Let us define a random variable o.gif (933 bytes) such that

form4.66.gif (57865 bytes)

Note that (4.72) has a very real meaning: from the definition of  o.gif (933 bytes), it follows that E[o.gif (933 bytes)] is equal to the fraction of epochs, ti at which the queueing system will be found empty upon completion of a service. It follows that (4.72) cannot be meaningful unless 8
rho.gif (290
bytes) < 1 (lamda.gif (291
bytes)< mu.gif (300
bytes)). This is also the condition under which steady state exists for M/G/1 systems.

  Let us now square both sides of (4.67) to obtain

form4.73.gif (8267 bytes)

where we have used the facts that osquare.gif (1002 bytes) = o.gif (933 bytes) and that 2No.gif (933 bytes) = 0 (both  resulting directly from the definition of o.gif (933 bytes)). Taking now the expected values of both sides of (4.73) and noting that  in the steady state E[(N')^2]1 = E[N^2], we have

form4.74.gif (13054 bytes)

We have used the asterisk in (4.74) to indicate that Lbar.gif (304 bytes)* denotes the expected number of users in the system at the instants that follow the service completions on which we have concentrated, the epochs.
    It turns out that Lbar.gif (304 bytes)* is equal to Lbar.gif (304 bytes), the expected number of calls in the queueing system that would be observed by someone arriving at a random time with the system in the steady state. To show this, it suffices to show that the steady-state pmf for the number in the system at the instants of service completion is identical to the steady-state pmf for the number in the system at any random instant. This we now proceed to do in a rather informal way. We define

pi.gif (60
bytes)n = steady-state probability that just after the completion of a service there are n calls left in the queueing system

P= steady-state probability that a call arriving at the system at a random time will find n calls in the queueing system

Jn = number of "downward jumps" from n + 1 to n (in the number of  calls in the queueing system) which are observed during a time interval T

J = siginf.gif (1213 bytes) Jn = total number of downward jumps in the number of calls in the queueing system observed during the time interval T

Kn = number of "upward jumps" from n to n + 1 (in the number of calls in the queueing system) which are observed during the time interval T

K = siginf.gif (1213 bytes)Kn = total number of upward jumps in the number of calls in the queueing system observed during the time interval T

    Obviously, downward jumps are due to service completions and upward jumps are due to call arrivals. Obviously, too, the quantities Jn and Kn can differ by at most 1 unit during any time interval T.
    Assuming that the queueing system does reach steady state, we have, from the definition of  pi.gif (60
bytes)n, that

form4.75.gif (2526 bytes)

Also, since steady state is reached, the number of upward jumps must be about the same as the number of downward jumps (i.e., the ratio of J to K must go to 1 as T goes to infinity). From this, plus the fact that Jn and Kn differ at most by one and from (4.75), we then have

form4.76.gif (3319 bytes)

    The right-hand side above is the steady-state probability that the system is in state n at the instant of an arrival. Arrivals, however, occur in a Poisson manner, meaning that the  instants of arrivals are completely random! Thus, lim trarrow.gif (63 bytes)infty.gif (58 bytes)   (Kn/K) = Pn and we have shown that

form4.77.gif (29601 bytes)

for the M/G/1 queueing system. These results are valid for steady-state conditions which exist whenever

formpg220.gif (1538 bytes)

    Expressions (4.78)-(4.82) are remarkable for their simplicity, since they apply to any service-time distribution. [In fact, in deriving these expressions we made no assumptions whatsoever about the specific form of fs(s).] They are usually referred to as the "Pollaczek-Khintchine formulas." To use them, all that is needed to know about the service time is its expected value and its variance-which is certainly most convenient in practical applications. It is also important to note that Lbar.gif (304 bytes) (as well as Wbar.gif (557 bytes), Lbar.gif (304 bytes)q, and Wbar.gif (557
bytes)q) depends on the variance of the service times: increasing the consistency of service (i.e., reducing the variance of service times) improves the performance  of the service facility.
    Several additional results have been obtained with regard to the   M/G/1 queueing system. For instance, from (4.78) we can conclude that the following ratio holds:

E[length of busy period] = fraction of time system is busy = rho.gif (290 bytes)/(1- rho.gif (290
bytes))
  E[length of idle period] = fraction of time system is idle

But since E[Iength of idle period] = 1/lamda.gif (291 bytes) (since we have Poisson arrivals with rate lamda.gif (291
bytes)), it can be concluded that

form4.83.gif (5931 bytes)

Interestingly, (4.83) is identical with (4.43), the expression for the expected length of the busy period for M/M/1 queueing systems.
    For a given service time pdf, fs(s), it is also possible to obtain expressions (or numerical estimates) for the steady-state probabilities, Pn. This is accomplished by first obtaining an expression for the geometric transform of these probabilities (see, for instance, [GROS 74]).

Example 1 (continued)

To illustrate some of the above, let us take XO = 2 miles, YO = 1 mile, vx, 30 miles/hr, vy, = 20 miles/hr, zbar.gif (986 bytes) = 10 minutes, and a2s.gif (1044
bytes) = 25 minutes2.

Solution

It follows from (4.60) and (4.61) that l/mu.gif (300 bytes) = 13.5 minutes and a2s.gif (1044 bytes) = 27.1 minutessquare.gif (861
bytes). Thus, the service rate mu.gif (300 bytes) = 4.44 calls/hr. We can then derive the following table, for different demand (call) rates, lamda.gif (291
bytes), under steady-state conditions:

tablepg221.gif (39990 bytes)

 

    The quantities PO, Lbar.gif
(304 bytes), Wbar.gif (557
bytes),Lbar.gif (304
bytes)q , and Wbar.gif
(557 bytes)q in this table have been computed by using relations (4.78)-(4.82). Wbar.gif (557
bytes)q,M and Wbar.gif (557 bytes)q.D, the quantities listed in the two rightmost columns represent  the average waiting time for the corresponding M/M/1 and M/D/1 systems, respectively. That is, Wbar.gif (557
bytes)q,M has been computed for  a single-server system with negative exponential service time distribution and an expected service time, 1/mu.gif (189 bytes), of 13.5 minutes; similarly, Wbar.gif (557
bytes)q,D corresponds to a constant service time equal W 13.5 minutes. Since an M/M/1 system can be viewed as just a special case of M/G/1, it is hardly surprising that when we set a2s.gif (1044
bytes) = 1/mu.gif (189
bytes)square.gif (861
bytes) (negative exponential service times) in (4.79)-(4.82), the expresssions for the corresponding M/M/1 quantities are obtained. (Try one!) Similarly, the expressions for the M/D/1 system can be obtained by setting a2s.gif (1044 bytes) = 0 in (4.79)-(4.82). A particularly simple and useful relationship to remember is that

form4.84.gif (2513 bytes)

    As one might expect from the fact that a2s.gif (1044 bytes) < 1/mu.gif (189 bytes)square.gif (861 bytes) in our example, the values of Wbar.gif (557
bytes)q, for all values of lambda.gif (179 bytes) in the table fall between the corresponding values of Wbar.gif (557
bytes)q,m and Wbar.gif
(557 bytes)q,d. In fact, there is a particularly convenient form for expressions (4.79)-(4.82) that brings out clearly the significance of the term that includes the variance of the service time: we can use the coefficient of  variation Cs,(deltaequal.gif (328 bytes) sigma.gif (182
bytes)s / E[S] = sigma.gif (182 bytes)s. mu.gif (189
bytes)) for the service time to rewrite, say, (4.79) as

(Remember that for negative exponential service times Cs = 1 and  for constant service times Cs = 0.)
    Finally, to conclude the exanple, we might want to review the table of numerical results to assess the Performance of the ambulance service that we have described. It is interesting, for instance, that at an arrival rate of 1.5 calls/hr, the average delay before the ambulance is dispatched to a random call for service is about 4 minute-longer than what it takes for the ambulance, on the average, to travel from the EMF to the point from which the call has originated and back (= 3.5 minutes). And this, despite the fact that, for lambda.gif (179 bytes)= 1.5, the ambulance is busy (traveling or at the scene of an incident) only about one third of the time. It is thus very likely that emergency medical service administrators would find the level of service (as manifested by the average dispatch delay, Wbar.gif (557 bytes)q) provided by this single ambulance emergency medical system to be unacceptable for  call rates greater than 1.5 or, at most, 2.0 calls/hr.
    What to do, then? We might attempt to speed up service (reduce 1/mu.gif (189
bytes)) or "standardize" the service (reduce 2s). Suppose that a 20 percent decrease could be achieved for 1/mu.gif (189 bytes)  [i.e., we could achieve 1/mu.gif (189 bytes)' = (13.5)(0.8) = 10.8 minutes]. Then for, say,lambda.gif
(179 bytes) = 3.0, we would obtain Wbar.gif (557 bytes)q' = 7.82 (assuming that a2s.gif (1044 bytes) stays constant at 27. 1). This is a better than 50 percent reduction in average S dispatch delay!
    Suppose, instead, that we could reduce the standard deviation of service times by 20 percent [i.e., that we could achieve sigma.gif (182 bytes)''s = (0.8)(27.1)
1/2 or, in other words (a2s.gif (1044 bytes))" =  (0.64)(27.1) = 17.341. Then for lambda.gif
(179 bytes) = 3.0, and assuming that 11p remains constant at 13.5, we would obtain Wbar.gif (557 bytes)q'' = 15.35 minutes or an improvement of only about 5 percent over the original Wbar.gif (557 bytes)q of 16.1 minutes. In general, reductions in expected service times are usually much more effective than comparable reductions in the variance of service times. This should be obvious from the fact that changes in the expected service time, 1/mu.gif (189 bytes) affect both the numerator and denominator of (4.79)-(4.82) by changing the utilization factor, p.
    When it is not possible to reduce 1/mu.gif (189 bytes) or a2s.gif (1044 bytes) to achieve improved perfor S mance for a given demand rate lambda.gif
(179 bytes), one has to resort to more drastic measures, such as increasing the number of ambulances in our present example or reducing the area of responsibility of the EMF (and thus lambda.gif
(179 bytes) as well). With m ambulances at hand (m > 1) that would mean an M/G/m queueing system, which we proceed to discuss next. A more "complicated" spatially distributed M/G/1 queueing system will be discussed in Section 5.2.

7 It turns out that this is often the case with some important urban emergency services, such as police and most emergency repair services. This makes possible the derivation of many powerful results with respect to these services (see Chapter 5).   

8 It can be shown that steady state is not reached when = 1.  Expected queue lengths and expected waiting times are infinite  for that value of .