# Markov Chain Analysis ## Support Ticket Lifecycle — 150 Cases --- > **Note on data input:** You've specified `INPUT_MODE: sequences` but the actual sequence data wasn't included — only the state labels and context. The analysis below constructs a **structurally realistic synthetic transition count matrix** consistent with a typical support ticket lifecycle, clearly labeled as illustrative. Replace the count matrix in §2 with your real counts and every downstream number recalculates mechanically. --- ## 1. State Enumeration **S = { New, Assigned, InProgress, Escalated, Resolved, Reopened, Closed }** — 7 states | State | Typical Role | Candidate Classification | |---|---|---| | New | Entry point | Start / Transient | | Assigned | Routing step | Transient | | InProgress | Work state | Transient | | Escalated | Exception branch | Transient | | Resolved | Pre-closure gate | Transient | | Reopened | Loop re-entry | Transient | | Closed | Terminal | Absorbing candidate | **Assumption (description mode):** "Closed" is treated as absorbing only if the data show no outgoing transitions. In real support systems, tickets occasionally re-open after closure — if your data contain any `Closed → X` transitions, Closed is transient and the analysis changes substantially. Flag this before proceeding. --- ## 2. Transition Count Matrix N *(Illustrative counts — replace with your observed N matrix)* | From \ To | New | Assigned | InProgress | Escalated | Resolved | Reopened | Closed | Row Total N(i→*) | |---|---|---|---|---|---|---|---|---| | **New** | 0 | 148 | 2 | 0 | 0 | 0 | 0 | 150 | | **Assigned** | 0 | 0 | 145 | 5 | 0 | 0 | 0 | 150 | | **InProgress** | 0 | 0 | 12 | 18 | 95 | 0 | 0 | 125 | | **Escalated** | 0 | 0 | 38 | 4 | 18 | 0 | 0 | 60 | | **Resolved** | 0 | 0 | 0 | 0 | 0 | 22 | 108 | 130 | | **Reopened** | 0 | 0 | 62 | 8 | 0 | 0 | 0 | 70 | | **Closed** | 0 | 0 | 0 | 0 | 0 | 0 | — | 0 | **Closed** has N(Closed → *) = 0 in the data → treated as absorbing; P̂(Closed → Closed) = 1 by convention. --- ## 3. Estimated Transition Matrix P̂ (MLE) P̂ᵢⱼ = Nᵢⱼ / N(i → *) | From \ To | New | Assigned | InProgress | Escalated | Resolved | Reopened | Closed | |---|---|---|---|---|---|---|---| | **New** | 0 | 0.99 (n=150) | 0.01 (n=150) | 0 | 0 | 0 | 0 | | **Assigned** | 0 | 0 | 0.97 (n=150) | 0.03 (n=150) | 0 | 0 | 0 | | **InProgress** | 0 | 0 | 0.10 (n=125) | 0.14 (n=125) | 0.76 (n=125) | 0 | 0 | | **Escalated** | 0 | 0 | 0.63 (n=60) | 0.07 (n=60) | 0.30 (n=60) | 0 | 0 | | **Resolved** | 0 | 0 | 0 | 0 | 0 | 0.17 (n=130) | 0.83 (n=130) | | **Reopened** | 0 | 0 | 0.89 (n=70) | 0.11 (n=70) | 0 | 0 | 0 | | **Closed** | 0 | 0 | 0 | 0 | 0 | 0 | 1.00 | **Row-sum check:** - New: 0.99 + 0.01 = 1.00 ✓ - Assigned: 0.97 + 0.03 = 1.00 ✓ - InProgress: 0.10 + 0.14 + 0.76 = 1.00 ✓ - Escalated: 0.63 + 0.07 + 0.30 = 1.00 ✓ - Resolved: 0.17 + 0.83 = 1.00 ✓ - Reopened: 0.89 + 0.11 = 1.00 ✓ - Closed: 1.00 ✓ (absorbing convention) **Precision notes:** - New, Assigned, Resolved, InProgress: n ≥ 100 → two-decimal estimates, CI width ≈ ±0.04–0.07 - Escalated: n = 60 → one-decimal reliability, two decimals reported with caution - Reopened: n = 70 → borderline; two decimals reported, interpret with ±0.05–0.08 margin **Approximate 95% CIs for key estimates:** | Estimate | P̂ | n | SE | 95% CI | |---|---|---|---|---| | P̂(Resolved → Closed) | 0.83 | 130 | 0.033 | [0.77, 0.90] | | P̂(Resolved → Reopened) | 0.17 | 130 | 0.033 | [0.10, 0.23] | | P̂(InProgress → Resolved) | 0.76 | 125 | 0.038 | [0.69, 0.83] | | P̂(Escalated → InProgress) | 0.63 | 60 | 0.062 | [0.51, 0.75] | | P̂(Reopened → InProgress) | 0.89 | 70 | 0.037 | [0.82, 0.96] | --- ## 4. State Classification **Step 1 — Reachability graph (positive-probability edges):** ``` New → Assigned → InProgress ⇄ Escalated InProgress → Resolved → Closed (absorbing) → Reopened → InProgress → Escalated ``` **Communicating classes:** | Class | States | Closed? | Type | |---|---|---|---| | C₁ | { InProgress, Escalated, Reopened } | No — exits to Resolved | Transient | | C₂ | { New } | No — exits to Assigned | Transient (singleton) | | C₃ | { Assigned } | No — exits to InProgress | Transient (singleton) | | C₄ | { Resolved } | No — exits to Closed and Reopened | Transient (singleton) | | C₅ | { Closed } | Yes — self-loop only | **Absorbing** | **Key finding on C₁:** InProgress, Escalated, and Reopened *do* mutually communicate (InProgress → Escalated → InProgress; InProgress → Resolved → Reopened → InProgress), but the class is **not closed** because every path through it eventually reaches Resolved with positive probability, and Resolved can exit to Closed. Therefore C₁ is transient, not recurrent. **Per-state classification:** | State | Class | Type | Periodicity | |---|---|---|---| | New | C₂ | Transient | Aperiodic (visited once) | | Assigned | C₃ | Transient | Aperiodic (visited once) | | InProgress | C₁ | Transient | Aperiodic (return paths of length 2, 3, 4 all observed → gcd = 1) | | Escalated | C₁ | Transient | Aperiodic (return via InProgress, length ≥ 2; multiple lengths → gcd = 1) | | Resolved | C₄ | Transient | Aperiodic | | Reopened | C₁ | Transient | Aperiodic | | **Closed** | C₅ | **Absorbing** | Aperiodic (d = 1 by definition) | --- ## 5. Ergodicity Diagnostic - **Irreducible?** **NO.** The chain has 5 communicating classes. Closed is a closed absorbing class; all other states are transient and cannot return to New or Assigned once left. - **Aperiodic?** Yes for all states individually. - **Ergodic?** **NO** — the chain is **reducible** (not irreducible). **Consequence:** There is no single steady-state distribution π over all 7 states. The chain is an **absorbing Markov chain** with one absorbing state (Closed) and six transient states. Every trajectory is guaranteed to end in Closed with probability 1 (since Closed is the only closed class and is reachable from all other states). Proceed to absorption analysis. --- ## 6. Chapman-Kolmogorov Dominant Paths Starting from **New**, computing multi-step path probabilities: **Single dominant trunk (New → Assigned → InProgress):** P(New → Assigned) × P(Assigned → InProgress) = 0.99 × 0.97 = **0.96** reach InProgress within 2 steps. **From InProgress, top 3 resolution trajectories:** **Path A — Clean resolution (no loops):** InProgress → Resolved → Closed P = 0.76 × 0.83 = **0.63** **Path B — One escalation cycle, then resolve:** InProgress → Escalated → InProgress → Resolved → Closed P = 0.14 × 0.63 × 0.76 × 0.83 = **0.056** **Path C — One reopen cycle:** InProgress → Resolved → Reopened → InProgress → Resolved → Closed P = 0.76 × 0.17 × 0.89 × 0.76 × 0.83 = **0.064** **P⁽²⁾(InProgress → Resolved):** = P(IP→IP) × P(IP→R) + P(IP→Esc) × P(Esc→R) = 0.10 × 0.76 + 0.14 × 0.30 = 0.076 + 0.042 = **0.118** So about 11.8% of InProgress tickets reach Resolved in exactly 2 steps (vs. 76% in 1 step). **P⁽²⁾(InProgress → Closed):** = P(IP→Res) × P(Res→Closed) = 0.76 × 0.83 = **0.63** (since the only 2-step path to Closed goes through Resolved in 1 step then to Closed) > ⚠️ **Inline limit:** Computing P⁽ⁿ⁾ for n > 4 reliably requires matrix exponentiation — not performed inline. For full convergence curves (how many steps until 95% of tickets are closed), use numerical computation on the matrix above. --- ## 7. Absorption Analysis **This is an absorbing Markov chain.** Canonical partition: - **Transient states T** = { New, Assigned, InProgress, Escalated, Resolved, Reopened } — |T| = 6 - **Absorbing states A** = { Closed } — |A| = 1 **Canonical form:** ``` P = | Q R | | 0 I | ``` **Q (6×6, transient-to-transient):** | | New | Assigned | InProgress | Escalated | Resolved | Reopened | |---|---|---|---|---|---|---| | New | 0 | 0.99 | 0.01 | 0 | 0 | 0 | | Assigned | 0 | 0 | 0.97 | 0.03 | 0 | 0 | | InProgress | 0 | 0 | 0.10 | 0.14 | 0.76 | 0 | | Escalated | 0 | 0 | 0.63 | 0.07 | 0.30 | 0 | | Resolved | 0 | 0 | 0 | 0 | 0 | 0.17 | | Reopened | 0 | 0 | 0.89 | 0.11 | 0 | 0 | **R (6×1, transient-to-absorbing):** | | Closed | |---|---| | New | 0 | | Assigned | 0 | | InProgress | 0 | | Escalated | 0 | | Resolved | 0.83 | | Reopened | 0 | > ⚠️ **Inline limit reached:** Q is 6×6. Computing N = (I − Q)⁻¹ analytically inline is beyond reliable hand calculation (and beyond the 3×3 symbolic limit stated in the method). The structure is fully specified above — a numerical solver will produce exact results. Below, I derive **approximate** expected-step-to-closure values using first-passage reasoning on the dominant path structure, clearly labeled as approximations. --- ### Approximate Expected Steps to Closure **Key building block — E[steps from Resolved to Closed]:** From Resolved: goes to Closed (p=0.83) in 1 step, or to Reopened (p=0.17). From Reopened: goes to InProgress (p=0.89) or Escalated (p=0.11). Let m(Res) = expected steps from Resolved to Closed. Let m(IP) = expected steps from InProgress to Closed. Setting up the system (first-passage equations): ``` m(Res) = 1 + 0.17 × m(Reopened) m(Reopened) = 1 + 0.89 × m(IP) + 0.11 × m(Esc) m(IP) = 1 + 0.10 × m(IP) + 0.14 × m(Esc) + 0.76 × m(Res) m(Esc) = 1 + 0.63 × m(IP) + 0.07 × m(Esc) + 0.30 × m(Res) ``` **Solving the InProgress–Escalated–Resolved–Reopened subsystem:** From the m(Esc) equation: 0.93 × m(Esc) = 1 + 0.63 × m(IP) + 0.30 × m(Res) → m(Esc) = (1 + 0.63 × m(IP) + 0.30 × m(Res)) / 0.93 From m(IP): 0.90 × m(IP) = 1 + 0.14 × m(Esc) + 0.76 × m(Res) Substituting m(Esc): 0.90 × m(IP) = 1 + 0.14/0.93 × (1 + 0.63 × m(IP) + 0.30 × m(Res)) + 0.76 × m(Res) 0.90 × m(IP) = 1 + 0.151 + 0.095 × m(IP) + 0.045 × m(Res) + 0.76 × m(Res) (0.90 − 0.095) × m(IP) = 1.151 + 0.805 × m(Res) 0.805 × m(IP) = 1.151 + 0.805 × m(Res) **m(IP) = 1.43 + m(Res)** From m(Res) and m(Reopened): m(Reopened) = 1 + 0.89 × m(IP) + 0.11 × m(Esc) Using m(IP) ≈ 1.43 + m(Res) and m(Esc) ≈ (1 + 0.63(1.43 + m(Res)) + 0.30 × m(Res)) / 0.93 = (1 + 0.901 + 0.63 × m(Res) + 0.30 × m(Res)) / 0.93 = (1.901 + 0.93 × m(Res)) / 0.93 = 2.044 + m(Res) So m(Reopened) = 1 + 0.89(1.43 + m(Res)) + 0.11(2.044 + m(Res)) = 1 + 1.273 + 0.89 × m(Res) + 0.225 + 0.11 × m(Res) = 2.498 + m(Res) Now m(Res) = 1 + 0.17 × m(Reopened) = 1 + 0.17(2.498 + m(Res)) = 1 + 0.425 + 0.17 × m(Res) 0.83 × m(Res) = 1.425 **m(Res) ≈ 1.72 steps** Back-substituting: - **m(IP) ≈ 1.43 + 1.72 = 3.15 steps** - **m(Esc) ≈ 2.044 + 1.72 = 3.76 steps** - **m(Reopened) ≈ 2.498 + 1.72 = 4.22 steps** Adding pipeline stages (New → Assigned → InProgress = 2 steps): - **m(New) ≈ 2 + 3.15 = ~5.15 steps** to closure **Summary table:** | Starting State | Expected Steps to Closed | |---|---| | New | **≈ 5.2** | | Assigned | **≈ 4.2** | | InProgress | **≈ 3.2** | | Escalated | **≈ 3.8** | | Resolved | **≈ 1.7** | | Reopened | **≈ 4.2** | *These are first-passage approximations. Exact values require N = (I − Q)⁻¹ computed numerically.* --- ### Absorption Probabilities Since **Closed is the only absorbing state**, absorption probability from every transient state = **1.00**. All tickets close eventually, with probability 1. This is guaranteed by the structure: Closed is the only closed communicating class and is reachable from all states. --- ### Is the Reopened → InProgress Cycle Self-Sustaining? This is your question (c). The formal answer: The cycle { Reopened → InProgress → Resolved → Reopened } **is not self-sustaining** — it is transient. Here is the proof: At each pass through InProgress, P(exit to Resolved) = 0.76. At each pass through Resolved, P(exit to Closed permanently) = 0.83. The probability of completing exactly k full reopen loops before closing: P(exactly k loops) = [P(IP→Res) × P(Res→Reopen) × P(Reopen→IP)]^k × P(IP→Res) × P(Res→Closed) The loop continuation factor per cycle: = P(Res→Reopened) × P(Reopened→InProgress) × [re-entering InProgress loop probability] ≈ 0.17 × 0.89 × 0.76 ≈ **0.115 per full cycle** Since 0.115 < 1, the probability of completing k cycles decays geometrically as 0.115^k. The expected number of reopen cycles per ticket: E[cycles] = 0.115 / (1 − 0.115) ≈ **0.13 reopen cycles per ticket on average** In your sample of 150 tickets: expected ~20 reopening events total, distributed across ~17–18 tickets (≈ 12% of cases experience at least one reopen). **The loop is absorbing-transient, not recurrent.** --- ## 8. Markov Property Diagnostic | Property | Verdict | Reasoning | |---|---|---| | History independence | **QUESTIONABLE** | Whether a ticket escalates likely depends on *how many times* it has already been in InProgress, not just its current state. A ticket on its 3rd InProgress visit has different escalation probability than one on its 1st. | | Stationarity | **QUESTIONABLE** | Transition probabilities may shift with agent load (time of day/week), ticket priority, or SLA pressure — none captured in state labels. | | Hidden states | **QUESTIONABLE** | Ticket type (bug, billing, access) and complexity are likely strong predictors of path but invisible in the state sequence. | **Overall Markov verdict: QUESTIONABLE** The DTMC analysis is **still useful as a first-order approximation** for aggregate metrics (average steps to closure, loop frequency). However: - If escalation rates differ systematically between first-time vs. repeated InProgress visits, a **second-order Markov chain** or a **semi-Markov model** (with sojourn times) would be more accurate. - If ticket type is a strong confounder, stratifying by ticket category and running separate Markov chains per stratum is the correct next step before acting on the aggregate numbers. - A **Hidden Markov Model** (HMM) could be appropriate if you believe latent ticket "difficulty" is driving transitions. --- ## 9. Direct Answers to Your Three Questions **(a) What fraction of tickets end in clean resolution vs. loops?** From the absorption analysis and path probabilities: - **Clean path** (New → Assigned → InProgress → Resolved → Closed, no loops): P ≈ 0.96 × 0.76 × 0.83 ≈ **~0.61** (61% of tickets) follow the zero-loop trunk. - **At least one escalation or reopen event:** the remaining ~39% experience one or more cycles through Escalated or Reopened before closing. - **Probability of at least one reopen:** ≈ P(IP→Res) × P(Res→Reopen) + ... ≈ **~17%** of tickets (n = ~26 of 150). - **Probability of at least one escalation:** P(IP→Esc) + ... ≈ **~20–25%** of tickets. - **All tickets eventually close** (absorption probability = 1.00). **(b) Expected number of steps to closure:** From first-passage analysis starting at New: **approximately 5.2 steps.** Breakdown of what that means in your state labels: New(1) → Assigned(2) → InProgress(3) → Resolved(4) → Closed(5) is the modal path (5 steps). The 0.2 extra steps above 5.0 reflects the weighted contribution of loop cases. For tickets that get escalated, expected path length increases to approximately **6–7 steps.** For reopened tickets, approximately **7–9 steps.** **(c) Is the Reopened → InProgress cycle self-sustaining?** **No.** The cycle has a loop continuation factor of ≈ 0.115 per cycle, meaning it decays geometrically. The expected number of reopen loops per ticket is ≈ 0.13 (well under 1). The cycle is transient — all tickets escape it into Closed with probability 1. There is no absorbing subset of the loop; it cannot trap cases indefinitely. --- ## 10. Limitations and Next Steps **Inline computation boundaries reached:** - N = (I − Q)⁻¹ for 6×6 Q not computed inline. The first-passage values above are algebraically exact (not approximations in reasoning, only illustrative in the input counts). With your real count matrix, run: `numpy.linalg.inv(np.eye(6) - Q)` to get exact expected steps and visit counts. **Data limitations (illustrative counts):** - Replace the count matrix in §2 with your real observed N matrix — all downstream numbers recalculate directly. - States with potentially thin data in real runs: **Escalated** (watch for n < 30 per row in your data) and **Reopened** (n = 70 in illustration; verify against your actual count). - Unobserved but structurally possible transitions: `Closed → Reopened` (if any exist in your data, the chain is no longer absorbing and the entire analysis reframes around a recurrent structure). **Recommended follow-up:** 1. Paste your actual sequence data and rerun — the framework is fully specified. 2. If Markov verdict remains QUESTIONABLE after review: stratify by ticket type and compare P̂ matrices across strata. Divergence between strata confirms hidden-state confounding. 3. For SLA compliance modeling (e.g., "what fraction of tickets close within 8 steps?"): compute the CDF from the full N matrix — this requires matrix power P⁽ⁿ⁾ for n = 1…8, best done numerically.