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Recall that the QED operational regime is characterized by (4.4). It turns out that different patience distributions give rise to different asymptotic behavior of performance measures. Therefore, several special cases are considered in Theorems 15.1-15.7.

Main case: positive density at the origin. Assume the density of patience exists at the origin and denote its value by g0. In most applications that we have encountered,

a non-negligible abandonment rate during the first seconds of wait was observed. (See Section 19, for example.) Hence, there is a practical motivation to treat, as the main case, patience distributions with a positive density at the origin: g0 > 0. In addition,

there are significant theoretical reasons for this emphasis. It turns out that, in this case, performance measures behave similarly to Erlang-A, as analyzed in Garnett et al. [29]: the probability of wait converges to a constant, and the probability to abandon and average wait decrease at rate 1/√n. If wait is measured in units of average-service-time, convergence rates depend only on the ratio g0/µ and the service grade β. In fact, in order

to get the right asymptotic expressions, when g0 > 0, one simply replaces the exponential

abandonment rate θ in the formulae of [29] by g0, the patience density at the origin.

In addition, we establish an asymptotic linear relation between the probability to abandon and average wait. The exact relation

P{Ab} = θ · E[W ] ,

that holds for models with exponential patience, is replaced in Theorem 15.1 by P{Ab} ≈ g0· E[W ] .

Although the last relation is approximate, our numerical experiments (Subsection 15.2) show that it provides an excellent approximation for a wide range of model parameters.

Since the case g0 > 0 is the most important, we compute for it more performance mea-

sures than in the other special cases: for example, asymptotics for E[W |Ab], E[W |W > t] etc. are derived only for this case; see Theorem 15.1.

Density vanishing near the origin. We would like to cover models where customers are going through several stages of (im)patience before reneging. (See, for example,

Issaev [38] or Baccelli and Hebuterne [3] who fit an Erlang distribution with 3 phases to patience in real data.) In such models, we cannot expect significant abandonment near the origin, which suggests patience distributions with density vanishing near the origin. These distributions are analyzed in Theorem 15.2. Specifically, assume that the k-th derivative of the density is positive and the first (k − 1) derivatives are zero. Then, in contrast to Theorem 15.1, positive, negative or zero values of the service grade β give rise to different performance regimes.

• If β > 0, the wait characteristics behave similar to the Erlang-C queue, described in Halfin and Whitt [34]. The probability to abandon decreases at n−(k+1)/2 rate, i.e. faster than in the main case. Both statements above are connected: since abandonment is negligible, the system behaves like Erlang-C

• If β < 0, almost all customers are delayed and the average wait decreases to zero slowly (at rate n−1/(2k+2)). The probability to abandon is asymptotically −β/√n, which is the minimal abandonment that is required to avoid queue explosion. • The case β = 0 implies some intermediate behavior (e.g. the average wait decreases

at rate n−1/(k+2)).

An important special case of distributions, covered by Theorem 15.2, is phase-type (see Issaev [38] and references therein for their importance in Queueing Theory). Theorem

15.3 and formula (15.47) below show how to calculate the first non-zero derivative at the origin for these distributions.

Delayed distributions. Assume that, up to a fixed time c > 0, customers do not abandon. For example, customers could be listening to an announcement. Such situations inspire us to consider delayed distributions of patience, which can be represented by c + τ , where τ represents (im)patience as before. (Recall the first plot of Figure 8.) The case of deterministic patience is important as well. As examples, one can consider overflowing3_,

or Internet applications, where the waiting of jobs in queue is usually bounded.

Theorem 15.4 and 15.5 cover the two cases: delayed distributions and deterministic. Overall, our main conclusions are similar to Theorem 15.2; positive, negative and zero

3_{Customers that do not get service within a deterministic target time are sent to another call center}

values of β should be treated separately again. However, if β ≤ 0, the average wait does not converge to zero. For negative service grades, it converges to the delay constant c, and if β = 0, to some number within the interval (0, c), which we identify in Theorem

15.4.

Balking. Let customers that do not get service immediately balk with probability P{Blk}. From a practical point of view, this means that some customers do not agree to wait at all. (The reader surely recalls such a situation from personal experience.)

In this case (Theorem 15.6), the QED operational regime implies performance charac- teristics that are sometimes reminiscent of the Erlang-B analysis in Jagerman [44]. The probability of wait decreases at rate 1/√n and the average wait of delayed customers decreases at rate 1/n. Hence, the unconditional average wait changes at rate n−3/2.

If a customer is not served immediately, the probability to abandon converges to P{Blk}. (So abandonments after positive wait are rare.) Finally, the most surprising result arises for the unconditional probability to abandon. Asymptotically, it decreases at rate 1/√n and is equal to the blocking probability in Erlang-B, derived in [44].

Scaled balking In our last special case, we assume that the balking probability is scaled by pb/

√

n, where n is the number of agents. It is designed to study queues with large number of servers and small, but non-negligible, balking probability. The findings, summarized in Theorem 15.7, are similar to the main case (Theorem 15.1).

Numerical experiments. The quality of the approximations from Theorems15.1,15.2

and 15.4 is checked in Subsection 15.2. Four patience distributions, seven values of the service grade and moderate-to-large values of λ and n were used in our experiments. If the offered load is larger than 100, we observe a good-to-excellent fit between approximations and exact values for absolute majority of special cases considered. A good fit for smaller values of the offered load (we started with 10) is also common.